11 Handbook - 2 Motion PDF

Motion 1+2: Introduction to Motion Welcome to the study of physics! As young scientists you will be making measurements and observations, building theories, and testing models that help us to describe how our world works. A: The Gold Medal Race A sixteen year-old swimmer from Toronto, Penny Oleksiak...

Motion 1+2: Introduction to Motion Welcome to the study of physics! As young scientists you will be making measurements and observations, building theories, and testing models that help us to describe how our world works. A: The Gold Medal Race A sixteen year-old swimmer from Toronto, Penny Oleksiak, won a gold medal in the women’s 100-m freestyle swimming competition at the 2016 Rio Summer Olympics. Your teacher will show you the video of this exciting race. A team of sports scientists and coaches have helped Penny reach this extraordinary level of performance. And since Penny was only 16 years old when she won, they expect her to get even better! You are now a sports scientist and your job is to analyze Penny’s race performance and help her improve. Thanks to Ryan Atkison from the Canadian Sports Institute, Ontario for the data from Penny Oleksiak’s race. 1. Explain. Watch the video of her gold-medal winning race. After watching, Isaac says, “I want to make some distance and time measurements for her motion, but I don’t know how. Her arms are moving, her legs are moving … it all seems very complicated!” Albert says, “I have an idea to simplify things. Let’s assume Penny is just a blob moving through the water. No arms, no legs.” In what ways is Albert’s idea crazy? In what ways is it reasonable and helpful? Crazy: Reasonable: Models and Assumptions. To make sense of a complex world, we create a simplified scientific picture called a model. We call the objects in our model that we want to study the system. Every model is built out of assumptions, which are statements about the system that are not quite 100% correct, but are probably pretty close. It is often helpful to use the point particle assumption, where we model the system as a small blob of matter. This is a reasonable assumption if we are not interested in the details of the object’s size or its shape. 2. Describe. Watch the video again. We will break up the race into different intervals where she is swimming or moving differently. Provide a short description of how she swims or moves during each interval of the race. Interval Description (1) Underwater (2) Front Crawl (3) Turn Around (4) Front Crawl Intervals. A scientific model is more reliable (it will give better predictions) when it focuses on a specific interval of time. 3. Reason. As a coach and sport scientist, you want to focus your attention on the longest interval of Penny’s race. You are concerned that she might be speeding up or slowing down too much during this interval. Why are you concerned? 4. Reason. In the fourth interval of Penny’s race, we measure that it takes her 17 s to travel 40 m. Is it possible to use these measurements to determine whether Penny is moving at steady rate? Explain. 5. Explain. Below are two sets of data from two different swimming races. Which one of these is an example of constant speed? (One of these is from Penny’s race!) 19  Position in Pool 15 m 25 m 35 m 45 m Time 6.52 s 11.83 s 17.10 s 22.69 s Position in Pool 15 m 25 m 35 m 45 m Time 6.52 s 11.83 s 17.14 s 22.45 s Manager: For the next question, ask your group to discuss it but don’t write anything! Then move on. 6. Define. (as a class) We need a definition that will allow us to test whether Penny, or any other object, moves with a constant speed. Constant Speed. B: Testing a Claim – Constant Speed Should we believe everything we see on the internet? A motorized toy car, which we affectionately call the physics buggy, is sold with this description: “Equally appealing to students of all ages, this simple but powerful toy provides a visible source of uniform speed.” Your task is to design an experiment that will test the claim that the toy moves with a constant speed. Position. To describe the position of an object along a line we need to know the distance of the object from a reference point, or origin, on that line and what direction it is in. One direction along the line is chosen to be the positive x-direction and the other negative. This choice is the sign convention. Choose your sign convention such that the position measurements you make today will be positive. +x origin positive x-direction 1. Plan. Your group will use one physics buggy, a large measuring tape (share if necessary) and a stopwatch (your smartphone with lap timer). (a) Describe a simple experiment using position and time measurements that will allow you to decide whether the buggy moves with a constant speed. (b) Draw a simple picture of the experiment, including the origin, and illustrate the quantities you will measure. (c) Describe what each person will do. 20 *** Check your plan with your teacher*** 2. Measure. Find your equipment and conduct your experiment. Record your data below. Careful! The time, t(s) below describes the total time for the buggy to travel from the origin to that particular position. The split times are the times between each position measurement. Position, x (m) Split Times (s) Time, t (s) 3. Reason. There are many sources of uncertainty in your time measurements. Do you think your time measurements in this experiment are reliable to the nearest 0.01 s? What do you think is a reasonable amount of uncertainty in your time measurements? Write down one sample time measurement using measurement notation. 4. Evaluate. Examine your time data. Keep in mind the measurement uncertainties. Make a decision: was your buggy moving with a fairly constant velocity? Explain. Motion Diagram. A motion diagram is a sequence of dots that represents the position of an object after equal intervals of time. We draw these dots along an axis that shows the positive direction and use a small vertical line to indicate the origin. The scale of your diagram is not important, as long as it shows the right ideas. 5. Interpret. Below is a motion diagram for one student’s buggy. Explain how you can tell whether the speed of this buggy was constant. ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ +x Position, x (m) Time, t (s) 6. Represent. We want to look for patterns in the position of the buggy. Plot your data on a graph with position on the vertical axis. 21 7. Find a pattern. To create a model of the motion of the buggy, we need to look for a pattern in the data. Do you believe your data is best modelled by a curving pattern or a straight-line pattern? How well does your data fit a straight-line pattern? Line of Best-Fit. The purpose of a line of best fit is to highlight a pattern that we believe exists in the data. Real data always contains uncertainties that lead to scatter (wiggle) amongst the data points. A best-fit line helps to average out this scatter and uncertainty. Any useful calculations made from a graph should be based on the best-fit line and not on the data chart or individual data points. As a result, we never connect the dots in our graphs of data. 8. Represent. Draw a line of best-fit for your data. No single point is more important than any other! 9. Reason. Imagine an experiment with a second buggy that produces a similar graph, but with a steeper line of best-fit. What is different about the movement of the second buggy? Explain. 10. Represent and Calculate. You are familiar with the expression for slope from your math class. Replace the math class symbols with physics symbols from this graph. For example, on this graph there are no “y” symbols, the x-axis position is on the vertical axis instead. Use the physics symbol “v” to replace the math symbol “m”. Next, substitute the values from the graph including their units. Compute the final result. 𝑦2−𝑦1 physics graph expression: math class expression: 𝑚 = 𝑥2−𝑥1 11. Interpret. According to your slope calculation, what does your buggy do every second? Velocity. The slope of a position graph gives the object’s velocity. In the study of physics, velocity has a very special meaning that makes it different from speed. You will learn more about this soon! 12. Evaluate. Based on your graphical results how well do the advertised claims for the buggy hold up? C: Penny’s Gold Medal Race Now back to our regularly scheduled program. The graph below shows the position and time data for Penny during her gold-medal race. Note that the data begins a short while after the start of the race. 1. Interpret. According to the data in the graph, is her speed constant? Explain how you decide. 22 2. Find a Pattern. Draw a line of best fit that matches her data. Use physics symbols to construct an expression for the slope of the graph. Use this to calculate her velocity. 3. Summarize. In how many different ways was constant speed represented in this investigation? Explain. Homework: Constant Speed Name: 1. The image above shows the International Space Station as it travels between the moon and Earth on February 4, 2017, as photographed by astral photographer Thierry Legault. The camera took photos after equal intervals of time. Is the speed of the ISS constant or changing? Explain. + A + B + C 2. Reason. Three different physics buggies produce the motion diagrams shown to the right. Rank the speed of the three buggies from fastest to slowest. Explain your reasoning. A B Position Time Position Time (cm) (s) (cm) (s) 0 0 0 0 15 1 2 5 30 2 6 10 45 3 12 15 60 4 20 20 3. Reason. Two student groups collect data tracking the motion of different toy cars, shown in the two charts of data. Which chart represents the motion of a car with constant speed? Explain how you can tell. 23 4. Canadian swimmer Penny Oleksiak completed the first 50 m of her gold medal race in a time of 25.7 s. (a) Estimate. Without using a calculator, estimate her speed during the first 50 m of her race. Show your simplified calculation. (b) Solve. Find Penny’s speed. Carefully go through the explanation process for calculations and cross out each sub-heading when you have completed that part of the process. (For example: Describe purpose, complete equations,....) Explanation process Describe purpose, complete equations, substitutions with units, calculate result, final statement Student Self-Evaluation Complete, correct, clear, concise I carefully showed and crossed out all steps in the explanation process (Q#4). I wrote numbers with units and an appropriate number of significant digits. This work would be useful for any student to study from in the future. Homework: Position Graphs Name: 1. Emmy walks along an aisle in our physics classroom. A 2. Use the position-time graph to construct a motion motion diagram shows her changing position. Two diagram for Isaac’s trip along the hallway from the events are labeled: (1) her starting position and (2) her washroom towards our class. We will set the classroom final position. Use the motion diagram sketch the door as the origin. Label the start (1) and end of the trip position graph. (A sketch doesn’t worry about exact (2). values.) Label the events on the graph. 1 2 24 Position, x Time, t +x Position, x Time, t 1 2 +x 3. Albert and Marie (1) leave their classrooms, (2) pass eachother in the hall and (3) arrive at their washrooms. Position, x (m) south Time, t (s) 1 3 2 +x 3 1 Albert Marie +x Albert Marie (a) Describe their motion. Albert: Marie: (b) How do the positions of Albert and Marie compare at event 2? 25  (c) Draw a separate motion diagram for Albert and for Marie. (d) Which interval of the journey takes more time? Explain how you can tell. Student Self-Evaluation My responses use thoughtful, complete sentences and are very easy to read. My graph is carefully sketched and uses event numbers My motions diagrams are neat and include event numbers This work would be useful for any student to study from in the future. (e) Who travels faster? Explain how you can tell. (f) Who reaches their respective washroom first? 26 Motion 3: Interpreting Position Graphs Today you will learn how to draw and interpret position-time graphs. A: Interpreting Position Graphs A student will move in front of a motion detector (sensor). The origin is at the sensor and the direction away from the sensor is chosen as the positive x-direction. 1. Interpret. (as a class) According to your teacher’s set-up, which compass direction does the positive x-direction correspond to? 2. Observe, Represent, and Interpret (as a class). A student will move according to the descriptions below. Sketch the graph from the computer, draw a motion diagram, add event numbers to the graph and diagram, and interpret the graph. (a) Standing still, close to the sensor Type of Graph: Feature Value Interpretation Starting position Size of slope Time +x Position (b) Standing still, far from the sensor Type of Graph: Feature Value Interpretation Starting position Size of slope +x Time Position (c) Walking slowly in the positive Type of Graph: direction at a steady rate. Feature Value Interpretation Starting position Shape of graph Size of slope +x Sign of slope Time Position 27 (d) Walking quickly in the positive Type of Graph: direction at a steady rate. Feature Value Interpretation Starting position Shape of graph Size of slope +x Sign of slope Time Position (e) Walking slowly in the negative Type of Graph: direction at a steady rate. Feature Value Interpretation Starting position Shape of graph Size of slope +x Sign of slope Time Position 3. Find a Pattern. Describe the difference between the position graphs made by walking slowly and quickly. 4. Find a Pattern. Describe the difference between the position graphs made by walking towards and away from the sensor. 5. Interpret. A proper position graph from Penny’s gold medal race looks more like the one to the right. Divide her race into two time intervals. Interpret the meaning of the position graph for each interval. The positive direction is west. (1): (2): B: The Position Prediction Challenge Now for a challenge! From the description of a set of motions, can you predict a more complicated graph? 28  A person starts 1.0 m in front of the sensor and walks in the positive direction slowly and steadily for 6 seconds, stops for 3 seconds, and then walks in the negative direction quickly for 3 seconds. Manager: Each person should work on their own for any question labeled “individually”. When done, ask everyone can share their responses and make changes. 1. Predict. (individually) Use a dashed line to sketch your prediction for the position-time graph for this set of motions. 2. Test. (as a class) Use the computer and motion detector to test your predictions. Make changes to your graph if necessary. C: Graph Matching Now for the reverse! To the right is a position-time graph and your challenge is to determine the set of motions which created it. Remember which direction is positive! 1. Interpret. (individually) Study the graph to the right and write down a list of instructions that describe how to move like the motion in this graph. Use words like fast, slow, at rest, constant, and compass directions. 0-6 seconds: 6-9 seconds: 9-12 seconds: 12-15 seconds: 2. Test. (as a class) Observe the results from the computer. Explain any important differences between your predictions and the ones which worked for our “walker”. 29 D: Summary 1. Summarize what you have learned about interpreting position-time graphs. Interpretation of Position-Time Graphs Graphical Feature Physical Meaning steep slope shallow slope zero slope sign of slope 2. What, in addition to the speed, does the slope of a position-time graph tell us about the motion on an object? Velocity. We have made a very important observation. The slope of the position-time graph is telling us more than just a number (how fast). We can learn another important property of an object’s motion that speed does not tell us. This is such an important idea that we give the slope of a position-time graph a special, technical name – the velocity of an object. The velocity is much more than just the speed of an object as we shall see in our next lesson! Aren’t you glad you did all that slope work in gr. 9?! Homework: Defining Velocity Name: Albert walks along York Mills Rd. on his way to school. Three important events take place. The +x direction is west. Event 1: At 8:07 Albert leaves his home. Event 2: At 8:28 Albert realized he has dropped his phone somewhere along the way and immediately turns around. Event 3: At 8:43 Albert finds his phone on the ground with its screen cracked (no insurance). | | | | | | | | | -4 -3 -2 -1 0 1 2 3 4 units = kilometers + x [W] x1 ∙ x3 ∙ X2 ∙ 1. Represent. Draw a vector arrow that represents the displacement for each interval of Albert’s trip and label them Δ x12 and Δx23. 2. Calculate. Complete the chart below to describe the details of his motion in each interval of his trip. Interval 1-2 2-3 Displacemen Δx12 = x2 – x1 t expression 30  Time interval Δt12 = t2 – t1 expression Δx23 = Δx12 = Displacemen t calculation Direction of movement Time interval Δt12 = result ∆𝑥12 Velocity 𝑣12 = ∆𝑡12 = 3. Explain. Why is the sign of the velocity different in each interval of his trip? 4. Calculate. What is his displacement for the entire trip? (Hint: which events are the initial and final events for his whole trip?) Draw an arrow that represents this displacement on the axis above. 5. Represent. In a different trip, Emma begins at her house with a position x1. She drives in her car and experiences a displacement of Δx. She arrives at a store with a position x2. (a) Write an expression for her final position using symbols only. Student Self-Evaluation The symbols I use have proper event numbers I always include units with every quantity I write. (b) Her house is positioned 3 km north of the origin and she has a displacement of 8 km south. Use your expression to find the position of the store. Motion 4: Defining Velocity To help us describe motion carefully we have been measuring positions at different moments in time. Now we will put this together and come up with an important new physics idea. Events. An event is something that happens at a certain place and at a certain time. We can locate an event by describing where and when that event happens. In grade 11, we will use the position (x) to describe where something happens and the time (t) to describe when. Often, there is more than one event that we are interested in so we label the position and time values with an event number: x2 or t3. A: Changes in Position = Displacement 31 Our trusty friend Emmy is using a smartphone app that records the events during her trip to school. Event 1 is at 8:23 when she leaves her home and event 2 is at 8:47 when she arrives at school. We track her motion along a straight line that we will call the x-axis and note the positions of the two events with the symbols x1 and x2. + x [East] x1 ∙ x2 ∙ | | | | | | | | | -4 -3 -2 -1 0 1 2 3 4 units = kilometers 1. Interpret. What is the position of x1 and x2 relative to the origin? Write your answer two ways: mathematically, using a sign convention, and in words describing the direction. math: x1 = 2 km x2 = words: x1: 2 kilometres east of the origin x2: 2. Reason and Interpret. In what direction did Emmy move? Describe this mathematically and in words. Use a ruler to draw an arrow just above the axis from x1 to x2 to represent her movement. math: words: Displacement. The change in position of an object is called its displacement (Δx) and is found by subtracting the initial position from the final position: Δx = xf – xi. The Greek letter Δ (“delta”) means “change in” and always describes a final value minus an initial value. In your work, you will always replace the subscripts “f ” and “i ” with the appropriate event number. The displacement can be visually represented by an arrow pointing from the initial to the final position. Any quantity in physics that requires a direction to describe it is called a vector quantity. 3. Reason. Is position a vector quantity? Explain. (Hint: to describe Emmy’s position, do we need to mention a direction?) 4. Reason. In the example above with Emmy, which event is the “final” event and which event is the “initial”? Which event number should we substitute for the “f ” and which for the “i ” in the expression for the displacement (Δx = xf – xi)? 5. Calculate and Interpret. Calculate the displacement for Emmy’s trip. What is the interpretation of the result? Be sure to mention the number part and the sign of the result. A sample is provided below. Sample using different values: Δx = x2 – x1 = 6 km – (-2 km) = 8 km, Emmy moved 8 km east 32 6. Calculate and Represent. Emmy continues her trip. Calculate the displacement for the following example. Draw a displacement vector that represents the change in position. | | | | | | | | | -4 -3 -2 -1 0 1 2 3 4 units = kilometers +x x2 ∙ x3 ∙ B: Changes in Position and Time In a previous investigation, we have compared the position of the physics buggy with the amount of time taken. These two quantities can create an important ratio. When the velocity is constant (constant speed and direction), the velocity of an object is the ratio of the displacement between a pair of events and the time interval. In equal intervals of time, the object is displaced by equal amounts. 1. Reason. Write an algebraic equation for the velocity in terms of v, x, Δx, t and/or Δt. (Note: some of these quantities may not be necessary.) 2. Calculate. Consider the example with Emmy between events 1 and 2. What was her displacement? What was the interval of time? Now find her velocity. Provide an interpretation for the result (don’t forget the sign!). In physics, there is an important distinction between velocity and speed. Velocity includes a direction while speed does not. Velocity can be positive or negative, speed is always positive. For constant velocity, the speed is the magnitude (the number part) of the velocity: speed = |velocity|. There is also a similar distinction between displacement and distance. Displacement includes a direction while distance does not. A displacement can be positive or negative, while distance is always positive. For constant velocity, the distance is the magnitude of the displacement: distance = |displacement|. C: Velocity and Speed Your last challenge is to find the velocity of Penny from her position-time graph. The positive direction is east. Event 1 is the start of the race, event 2 is when she turns around, and event 3 is when she touches the wall to finish. 1. Calculate. What is Penny’s displacement during each half of the race? Use the appropriate symbols! 33 2. Calculate. Find her velocity during each half of her race. 3. Calculate. Find her speed during each half of the race. 34 Motion 5: Velocity-Time Graphs We can represent velocity graphically like we did with position. But be careful: each graph has different rules for its interpretation. A: The Velocity-Time Graph 1. Interpret. According to your teacher’s set-up, what compass direction corresponds to the positive x-direction? 2. Observe and Interpret. (as a class) A student walks slowly in the positive x-direction with a constant velocity. Observe a student and record the results from the computer. You may smooth out the jiggly data from the computer. Walking slowly in positive direction Type of graph: Feature Value Interpretation Sign of velocity values Size of velocity values Time Size of Slope - Velocity + 3. Explain. Isaac was asked to predict the shape of the previous velocity graph. He drew the graph to the right. Explain what he was thinking when making this prediction. Time Velocity + 4. Predict. (individually) Sketch your prediction for the four velocity-time graphs that corresponds to each situation described in the chart below and continued on the next page. Use a dashed line for your predictions. (a) Walking quickly at a steady rate in Type of graph: the positive direction Feature Value Interpretation Sign of velocity values Size of velocity values Time Size of Slope - Velocity + (b) Start 3 m away, walk quickly in the Type of graph: negative direction at a steady rate. Time Feature Value Interpretation - Velocity + Sign of velocity values Size of velocity values Size of Slope 35 (c) Start 3 m away, walk slowly in the Type of graph: negative direction at a steady rate. Feature Value Interpretation Sign of velocity values Size of velocity values Size of Slope Time - Velocity + (d) Start 1.5 m away, walk slowly in Type of graph: the negative direction at a steady rate. Feature Value Interpretation Time - Velocity + Sign of velocity values Size of velocity values Size of Slope 5. Observe and Interpret. (as a class) The computer will display its results for each situation. Draw the results with a solid line on the graphs above. Remember that we want to smooth out the bumps and jiggles from the data. Complete the interpretation part of the chart. 6. Explain. Based on your observations of the graphs above, how is speed represented on a velocity graph? (How can you tell if the object is moving fast or slow)? 7. Explain. Based on your observations of the graphs above, how is direction represented on a velocity graph? (How can you tell if the object is moving in the positive or negative direction)? 8. Explain. If everything else is the same, what effect does the starting position have on a velocity graph? B: The Main Event! A person moves in front of a sensor. There are four events: (1) The person starts to walk slowly in the positive direction, (2) at 6 seconds the person stops, (3) at 9 seconds the person walks in the negative direction twice as fast as before, (4) at 12 seconds the person stops. 1. Predict. (individually) Use a dashed line to draw your prediction for the shape of the velocity-time graph for the motion described above. Label the events. 36 Velocity is a vector quantity since it has a magnitude (number) and direction. All vectors can be represented as arrows. In the case of velocity, the arrow does not show the initial and final positions of the object. Instead it shows the object’s speed and direction. 2. Represent. In the previous example, the person walked at 0.8 m/s during the first interval and at -1.6 m/s during the third interval. Define the right-hand direction on your page as positive. The two vector arrows drawn below represent these two velocities. Label each arrow with its velocity value. Explain how you decided. Homework: Velocity Graphs Name: 1. Two motion diagrams track the movement of a student walking in a straight line. position time (s) - velocity + time (s) + x [W] 1 3 2 position Time (s) - velocity + time (s) 3 + x [W] 37  1 2 (a) Represent. Sketch a position-time graph for each motion diagram. The scale along the position axis is not important. Use one grid line = 1 second for the time axis. Include event numbers. (b) Represent. Sketch a velocity-time graph for each motion diagram. The scale along the velocity axis is not important. (c) Interpret. Label each section of each representation as “fast” or “slow”. Is each set consistent? 2. The two graphs below show data from Penny Oleksiak’s 100-m gold-medal race. (a) Read. What is Penny’s speed at 22 s? What is her velocity at 22 s? (b) Read. What is Penny’s speed at 33 s? What is her velocity at 33 s? (c) Interpret. Is Penny’s speed constant? What about her velocity? What is your evidence? 38 Homework: Conversions Name: 1. You are driving in the United States where the speed limits are marked in strange, foreign units. One sign reads 65 mph which should technically be written as 65 mi/h. You look at the speedometer of your Canadian car which reads 107 km/h. Are you breaking the speed limit? (1 mi = 1.60934 km) 2. A car travels at 104 km/h. You want to know how much time it will take to pass through a tunnel that is 12.7 m long. You have two options: convert the speed to m/s or convert the distance to km. One will produce a result that is easy to interpret and the other a result that is harder to interpret. Choose the easy way and complete the calculation. 3. You are working on a nice muffin recipe only to discover, halfway through your work, that the quantity of oil is listed in mL. You only have teaspoons and tablespoons to use (1 tsp = 4.92 mL, 1 tbsp = 14.79 mL). Which measure is best to use and how many? 4. Your kitchen scale has broken down just as you were trying to measure the cake flour for your muffin recipe. Now all you have is your measuring cup. You quickly look up that 1 kg of flour has a volume of 8.005 cups. How many cups should you put in your recipe? Student Self-Evaluation Check-list My responses use thoughtful, complete sentences and are very easy to read. I assigned a symbol to each quantity I converted. I used a conversion ratio in each response. 39 I carefully showed how the units divide away in each response. This work would be useful for any student to study from in the future. 5. You convert a time interval from hours into years. Do you expect the number part to be a larger or smaller value? Explain. Motion 6: Conversions In our daily life we often encounter different units that describe the same thing – speed is a good example of this. Imagine we measure a car’s speed and our radar gun says “100 km/h” or “62.5 miles per hour”. The numbers (100 compared with 62.5) might be different, but the measurements still describe the same amount of some quantity, which in this case, is speed. A: The Meaning of Conversions When we say that something is 3 m long, what do we really mean? 1. Explain. “3 metres” or “3 m” is a shorthand way of describing a quantity using a mathematical calculation. You may not have thought about this before, but there is a mathematical operation (+, -, ×, ÷) between the “3” and the “m”. Which one is it? Explain. S.I. Units. Scientists uses a standard set of units, called S. I. (Système internationale) units, which are not always the ones used in day-to-day life. The S. I. units for distance and time are metres (m) and seconds (s). It is an important skill to be able to change between commonly used units and S.I. units. 2. Reason. Albert measures a weight to be 0.454 kg. He does a conversion calculation and finds a result of 1.00 lbs. He places a 0.454 kg weight on one side of a balance scale and a 1.00 lb weight on the other side. What will happen to the balance when it is released? Explain what this tells us physically about the two quantities 0.454 kg and 1.00 lbs. 3. Reason. There is one number we can multiply a measurement by without changing the size of the physical quantity it represents. What is that number? Conversion Ratio. A conversion of units leaves the physical quantity unchanged and only changes the number and unit being used to write the quantity. To make sure we don’t change the actual physical quantity when converting, we only ever multiply the measurement by a conversion ratio that always equals 1. m = 0.454 kg ( 2.204 𝑙𝑏𝑠 1 𝑘𝑔 ) = 1 lb v = 65 𝑘𝑚 ℎ ( 1 ℎ 3600 𝑠 )= 0.0180 km/s The ratio in the brackets is the conversion ratio. Note that the numerator and denominator are equal, making the ratio equal to “1”. 4. Explain. Two students are converting 3 km into meters. They each set up their math work using a conversion ratio. Whose work do you agree with? Explain. 40  Albert: ∆𝑥 = 3 𝑘𝑚 ( 1 𝑘𝑚 1000 𝑚 ) Isaac: ∆𝑥 = 3 𝑘𝑚 ( 1000 𝑚 1 𝑘𝑚 ) 5. Reason. You want to convert 16 minutes into seconds. Construct the mathematical expression to do this and explain why it will work. Follow the model below. symbol = original quantity with units x ( 𝑐𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 ) = result 41 B: The Practice of Conversions 1. Solve. Convert the following quantities. Carefully show your conversion ratios and how the units divide out. Remember to use our guidelines for significant digits! Convert to seconds Convert to 4.5 m to kilometers ∆x = ∆t = 12.5 minutes = 2. Reason. In the previous question, you converted from minutes to seconds. Explain in a simple way why it makes sense that the quantity measured in seconds is a bigger number. 3. Reason. You are converting a quantity from kilograms into pounds. Do you expect the number part to get larger or smaller? Explain. 4. Solve. Convert the following quantities. Carefully show your conversion ratios and how the units divide out. Don’t forget our guidelines for significant digits! Convert to 138 lbs to kilograms Convert to seconds ∆t = 3.0 days = 5. Reason. You are converting a quantity from km/h into m/s. How many conversion ratios will you need to use? Explain. Convert to m/s Convert Penny’s finishing race speed of 1.98 m/s to km/h. v = 105 = 6. Conversion Challenge. Choose an interesting object that belongs to your group. Your teacher has a collection of small weights. Your challenge is to assemble a group of weights that has the same mass as your interesting object. The trick is, the collection of weights are all measured in grams, and the digital balance scale only measures in ounces! (1 pound = 16 ounces) Go! When you are ready, test your result for your teacher. A good result will agree with the original measurement to within ± 0.1 oz. 42 Motion 7: Modeling Solutions to Problems Creating a model of a system and using it to make predictions requires thought and care. In our physics course, we do this using a five-part process. Let’s return to Penny`s gold-medal race and explore an example of this process. Manager: Help your group read carefully through this example. Members can take turns reading. Don’t skip anything! Problem: Penny dives into the pool and reaches the surface after swimming under water for 6.52 s. Then she swims the remaining 85 m of the race with a steady speed of 1.808 m/s. According to this model, what is her time for the entire race? A: Pictorial Representation (of Model) Sketch showing events, describe events, coordinate system, label givens & unknowns with symbols, conversions Event 1 = she reaches surface Event 2 = she reaches the finish t1 = 6.52 s v = 1.808 m/s t2 = ? Δx = 85 m ∆t = ? + x [East] B: Physics Representation (of Model) Motion diagram, motion graphs, velocity vectors, event numbers v t x t 1 2 2 1 + x [E] 43 v1 v2 1 2 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ C: Word Representation (of Model) Describe motion (no numbers), assumptions, estimated result (no calculations) She swims east (the positive direction). We assume Penny is a point particle and has a constant velocity. I estimate it will take her about 40 seconds to swim this distance. D: Mathematical Representation (of Model) Describe purpose of math, complete equations, algebraically isolate, substitutions with units, final statement of prediction Find Penny’s time to swim the remaining distance: v = Δx/Δt ∴Δt = Δx/v = (85 m)/(1.808 m/s) = 47.01 s Find the time for her entire race: ∆t = t2 – t1 ∴ t2 = t1 + ∆t = 6.52 s + 47.01 s = 53.53 s According to this model, I predict her race time will be 53.5 s. 1. Explain. Why do the given and unknown quantities have these locations relative to the sketch in this example? 44 2. Explain. Imagine you could only see part A of the solution. Explain how to decide whether any conversions are needed and should be done in part A. 3. Describe. Event 1 in this problem does not occur at the origin for position measurements. There are three ways that this is shown in the solution. Describe these three ways. Consistency. When we solve a problem using this solution process, we can check the quality of our solution by looking for consistency. For example, if the object is moving with a constant velocity we should see that reflected in many parts of the solution. If the object is moving in the positive direction, we should see that reflected in many parts. Always check to see that the important physics ideas are properly reflected in all parts of the solution. 4. Interpret. Our swimmer in this problem is swimming in the positive direction. List all the ways this is shown in the solution. 5. Interpret. Penny swims with a constant velocity. List all the ways this is shown in the solution. Algebraically Isolate. A new step in the explanation process for calculations (the Mathematical Representation) is the step “algebraically isolate”. Before we substitute any numbers into an equation, we will isolate the unknown variable on one side of the equation using symbols. Exercise those algebra skills you have worked so hard on in math class! 6. Explain. Carefully show all the mathematical steps used to rearrange the velocity equation to solve for time. Make sure you show how quantities divide away. (Note: the work shown in the sample solution is all you need to do in the future) ∆𝑥 𝑣= ∆𝑡 B: Problems Unsolved Use the new process to model a solution for the following problems. Use the blank solution sheet on the next page. To conserve paper, some people divide each page down the centre and do two problems on one page. 1. In a record-breaking race, Usain Bolt took 50 m to reach his top speed. After that, he ran the next 150 m of his race in 13.59 s. What is his speed in m/s and km/h during the last part of his race? The next problem involves vertical motion. Draw your sketch vertically and use the symbol y instead of x for the position. For this problem, choose upwards as the +y direction and the ground as the origin. 2. In February 2013, a meteorite streaked through the sky over Russia. A fragment broke off 35 km above the surface of the earth and traveled downwards with a velocity of -12 000 km/h. It exploded 10 s after breaking off. How far above the earth was the meteorite when it exploded? (Hint: set y1 = 35 km and watch for the negative velocity!) 45 Motion Solution Sheet Name: Problem: A: Pictorial Representation Sketch showing events, describe events, coordinate system, label givens & unknowns using symbols, conversions B: Physics Representation Motion diagram, motion graphs, velocity vectors, event numbers C: Word Representation Describe motion (no numbers), assumptions, estimated result (no calculations) D: Mathematical Representation Describe purpose of math, complete equations, algebraically isolate, substitutions with units, final statement of prediction 46 Homework: Representations of Motion Each column in the chart below shows five representations of one motion. The small numbers represent the events. Here are some hints for the motion diagrams: (a) If the object remains at rest, the two events will be located at the same position on a motion diagram (see situation 1), (b) if it changes direction, shift the dots just above or below the axis (see situation 1), (c) remember that the origin is marked by a small vertical line. There is at least one completed example of each type of representation that you can use as a guide. The positive direction is shown in the position graph and motion diagram. Situation 1 Situation 2 Situation 3 Situation 4 Description Description Description Description 1-2: 1-2: 1-2: 1-2: It starts at the origin and remains at rest for a while. 2-3: 2-3: 2-3: 2-3: It moves quickly in the positive direction (east) with a constant velocity 3-4: 3-4: 3-4: 3-4: It moves slowly in the negative direction (west) with a constant velocity. Position Graph Position Graph Position Graph Position Graph x x x x [W] [N] [S] [E] t t t t ∙1 1 2 3 4 ∙ ∙ ∙ ∙ Velocity Graph Velocity Graph Velocity Graph Velocity Graph 47 v v v v t t t t 1 2 3 4 Motion Diagram Motion Diagram Motion Diagram Motion Diagram + x [W] + x [N] + x [S] + x [E] 1 2 4 3 Velocity Vectors Velocity Vectors Velocity Vectors Velocity Vectors (velocity during each interval) (velocity during each interval) (velocity during each interval) (velocity during each interval) 1-2: 1-2: 1-2: 2-3: 1-2: 2-3: 2-3: 3-4: 2-3: 3-4: 3-4: 3-4: Motion 8+9: Changing Velocity We have explored the idea of velocity and now we are ready to test it carefully and see how far this idea goes. As you work through this investigation remember how we have interpreted the velocity ratio ∆x/∆t so far: “The quantity ∆x/∆t tells us how far and in what direction the object travels every second. For example: -3m/s means that for every second that goes by, the object travels 3 metres in the negative direction.” 48 A: Motion with Changing Velocity Your teacher has a tickertape timer, a cart and an incline set-up. Turn on the timer and then release the cart to run down the incline. Bring the tickertape back to your table to analyze. 1. Observe. Examine the pattern of dots on your tickertape. How can you tell whether or not the velocity of the cart was constant? 2. Find a Pattern. From the first dot on your tickertape, draw lines that divide the dot pattern into intervals of six spaces as shown below. Do this for 10 intervals. ∙∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ 0 1 2 3. Reason. The timer is constructed so that it hits the tape 60 times every second. How much time does each six-space interval take? Explain your reasoning. 4. Reason. Albert makes a calculation of the velocity ratio ∆x/∆t for the interval of his entire dot pattern on the ticker tape. He says, “My result is 53 cm/s. This means that for every second that goes by, the cart moves 53 cm in the positive direction.” Is Albert’s interpretation of this ratio reasonable? Explain. Average Velocity. When the velocity is noticeably changing during a time interval ∆t, we cannot use our simple interpretation of the ratio ∆x/∆t. Instead, we call the ratio ∆x/∆t the average velocity. This is our first hint that changing velocity is a very different state of motion than constant velocity. We need to develop a more powerful interpretation for the ratio ∆x/∆t in this new state of motion. B: Analyzing Motion with Changing Velocity On the next page are a chart for your position-time data and a grid for your graph. Follow the instructions below. 1. Measure. Collect a complete set of position and time data from your tickertape. Each position measurement should start from the first mark “0” you make. Record your data in the table on the next page. 2. Reason. What is the uncertainty in your position measurements? 3. Find a Pattern. Plot the data in a graph of position vs. time. Does the data seem to follow a straight-line pattern or a curve? Explain. Time, t (s) Position, x (cm) 0 49 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 4. Explain. We will focus on the time interval from 0 to 1.0 seconds. During this time interval: Position, x (cm) 0 5 10 15 20 25 30 35 40 45 50 55 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time, t (s) (a) How do the spaces between the ticker tape dots show that the velocity is changing? (b) How does the position data in the table above show that the velocity is changing? (c) Draw a straight line on your position graph that connects with your smooth curve at 0 and 1.0 s (this line is called a secant). How can you tell that the velocity during this time interval is not constant? 5. Explain. We will focus on a smaller interval of time from 0.4 to 0.6 s. Highlight this on your ticker tape. Imagine these are all the dots all you can see (the rest of the tape was hidden). (a) When you examine this interval of ticker-tape dots, can you tell if the velocity is changing? How? (b) Draw a secant (a straight line) on your position graph that connects with your smooth curve at 0.4 and 0.6 seconds. If you could only see this interval (the rest was hidden), can you tell if the velocity is changing? How? 6. Explain. Now let’s explore a very small interval, starting one single dot before 0.5 s and one single dot after 0.5 s. Highlight this on your tickertape. Imagine these are all the dots you can see. Can you tell if the velocity is changing? Tangent. As the time interval becomes smaller, the velocity within that interval appears more and more like constant velocity. We can always make the time interval smaller and smaller, and when we do that something remarkable happens: the line we have been drawing on our graph no longer appears to connect with the graph at two separate points. So we say that the line now touches the graph at one point. This type of line is called a tangent: a straight line that touches a curve at only one point without crossing over the curve. A B C 7. Apply. Use the new definition of a tangent to explain which lines are tangent to the curve shown to the right and which are not. 50 8. Represent and Reason. Draw a tangent to the smooth curve of our graph such that it touches the curve at 0.5 s. (a) Now hold your ruler against your curve as if you were going to draw another tangent at 0.4s (don’t draw it!). How does the slope of the tangent at 0.4 s compare with the one at 0.5 s? (b) Do the same for 0.6 s. How does this slope compare with 0.5 s? (c) What quantity or characteristic of the cart’s motion is changing each at every moment in time? (Hint: what does the slope of a position-time graph represent?) Instantaneous Velocity. The slope of the tangent to a curving position-time graph give the object’s instantaneous velocity: its velocity at one instant in time. This new type of velocity will be our focus for most of our work in physics. Since instantaneous velocities are different at different moments in time, it is helpful to label them with a subscript number that corresponds to the event. For example: v1 = 12 m/s means the instantaneous velocity at moment 1 (the time of event 1). 9. Calculate. Label the point on your curve at 0.5 s event “1”. Find the slope of the tangent to your curve. Hint: when you find the slope of any line, you will get a more reliable result (with less uncertainty) if you choose two points on the line that are far apart. Be sure to use physics symbols and show your units when you substitute the values. The slope of a tangent is the same as the slope of the curve at that point along the curve. It might seem strange to think of a curve as having a slope, since it has no section that is straight. But we use a trick: the tangent allows us to imagine what the graph might look like if the curve stopped curving! So, the slope of the tangent equals the slope of the curve at that point along the curve. The curve “stops curving”. At moment 1 we imagine the line follows the tangent. ∙ 1 A tangent to the curve at moment 1. ∙ 1 This provides an important clue to help us interpret the meaning of the ratio ∆x/∆t for an instantaneous velocity: it tells how the object would move if the velocity stopped changing. For example: v1 = +2 m/s means “At 0.5 seconds (the time of event 1), the object would travel 2 m in the positive direction every second if its velocity stopped changing.” 10. Interpret. Use the description above to interpret the slope result for your tangent. C: Summary of Velocity Ideas We started this lesson with a simple idea of velocity but have now learned that when velocity is changing, we need to refine our ideas in order to properly interpret the ratio ∆x/∆t. Here is a summary of what we have figured out. 51 The Meaning of the Velocity Ratio ∆x/∆t State of Motion Time Interval Interpretation of Ratio Label for Ratio Constant large “for every time interval Δt, the object will travel through a velocity, v Velocity displacement Δx” Changing large When the velocity noticeably changes: average Velocity “during the time interval Δt, the object travels through a velocity, vavg displacement Δx” small (or just When the velocity appears constant: instantaneous one moment “for every time interval Δt, the object would travel through a velocity, v1, v2, in time) displacement Δx if its velocity stopped changing” etc. The Language of Velocity. (1) In the future, most of our physics work will focus on situations with changing velocity. As a result, we tend to get lazy and just say “velocity” when we really mean instantaneous velocity. You can always decide which velocity we mean by thinking about the state of motion and the time interval involved. (2) The magnitude (the number part without the direction) of the instantaneous velocity is the instantaneous speed. We will often use the word speed to refer to the size of the instantaneous velocity. Homework: Changing Velocity– Do This Now! 1 ∙ Position, x (cm) east 0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Time, t (s) 1. Calculate. Find the slope of the tangent to the curve at moment 1. This represents the instantaneous velocity at what moment in time? 2. Interpret. Draw three other tangents at different places on the graph. (a) Label them “faster” or “slower” compared with the tangent that was drawn for you. (b) Label the directions represented by the slopes of your tangents “east” or “west”. 3. Explain. Is it possible to draw a tangent to this graph that represents an instantaneous velocity of zero? Explain and, if possible, draw. 4. Explain. At the beginning of her race, Penny Oleksiak dove into the water and swam under water for a 10-m distance. During this time, she was slowing down and at one moment had a velocity of 1.99 m/s. Explain how to use the summary chart above to decide what type of velocity this is. 5. Interpret. Google Maps told you that your drive trip to school involved traveling 3 km north and took 5 minutes. You use these values in a calculation of the velocity ratio. Explain how to use the summary chart above to decide what type of velocity this is. 52 Motion 10: The Idea of Acceleration A: The Idea of Acceleration Interpretations are powerful tools for making sense of calculations. Please answer the following questions by thinking and explaining your reasoning to your group, rather than by plugging into equations. Consider the situation described below: A car was traveling with a constant velocity 20 km/h south. The driver presses the gas pedal and the car begins to speed up at a steady rate. The driver notices that it takes 3 seconds to speed up from 20 km/h to 50 km/h. 1. Interpret. A student who is studying this motion subtracts 50 – 20, obtaining 30. How would you interpret the number 30? What are its units? 2. Interpret. Next, the student divides 30 by 3 to get 10. How would you interpret the number 10? (Warning: don’t use the word acceleration, instead explain what the 10 describes a change in. What are the units?) 3. Reason. How fast is the car going 2 seconds after starting to speed up? Explain (don’t do a calculation). 4. Reason. How much time would it take to go from 20 km/h to 80 km/h? Explain (don’t do a calculation). B: Watch Your Speed! Shown below are a series of images of a speedometer in a car showing speeds in km/h. Along with each is a clock showing the time (hh:mm:ss). Use these to answer the questions regarding the car’s motion. 1. Reason. What type of velocity (or speed) is shown on a speedometer – average or instantaneous? Explain. 2. Explain. Is the car speeding up or slowing down? Is the change in velocity steady? 53 3. Explain and Calculate. Explain how you could find the acceleration of the car. Calculate this value and write the units as (km/h)/s. 4. Interpret. Albert exclaims, “In our previous result, why are there two different time units: hours and seconds? This is strange!” Explain to Albert the significance of the hours unit and the seconds unit. The brackets provide a hint. C: Interpreting Velocity Graphs To the right is the velocity versus time graph for a car. Two moments, 1 and 2, are indicated on the graph. time, t (seconds) 5 10 15 0 1 2 3 4 5 6 7 8 9 100 0 1 2 c d velocity, v (m/s) 1. Interpret. What information does point 1 tell us about the car? What about point 2? 54 2. Interpret. Give an interpretation of the interval labelled c. What symbol should be used to represent this? 3. Interpret. Give an interpretation of the interval labelled d. What symbol should be used to represent this? 4. Interpret. Give an interpretation of the ratio d/c. How is this related to our discussion in part A? 5. Calculate. Calculate the ratio d/c including units. Write the units in a similar way to question B#3. 6. Explain. Use your grade 8 knowledge of fractions to explain how the units of (m/s)/s are simplified. Motion 11: Calculating Acceleration Acceleration. The slope of a velocity graph is the acceleration of the object. When the velocity of an object changes by the same amount in equal time intervals, it experiences constant acceleration. In that case, we can interpret the ratio Δv/Δt as the change in velocity occurring in each equal interval of time. The ratio, Δv/Δt, is the definition of acceleration and is represented by the symbol, a. a = Δv/Δt = , if the acceleration is constant In Gr. 11 physics, we will focus on situations in which the acceleration is constant (sometimes called uniform acceleration). Acceleration can mean speeding up, slowing down, or a change in an object’s direction - any change in the velocity qualifies! In the equation above, we wrote vf and vi for the final and initial velocities during some interval of time. If your time interval is defined by events 2 and 3, you would write v3 and v2 for your final and initial velocities. 1. Explain. We mentioned earlier that the “Δ” symbol is a short form. In this case, explain carefully what Δv represents using both words and symbols. 55 2. Solve. Your teacher has a cart set up on a track at the front of the room. The cart is equipped with a fan that causes it to speed up. Your teacher will release the cart and use the motion detector to produce a graph of its velocity. What is the cart’s acceleration? (You can check your answer by using the computer to find the slope of the velocity graph.) A: Pictorial Representation B: Physics Representation Sketch showing events, describe events, coordinate system, label givens & Motion diagram, velocity graph, velocity vectors, event numbers unknowns using symbols, conversions C: Word Representation Describe motion (no numbers), assumptions, estimated result (no calculations) D: Mathematical Representation Describe purpose of math, complete equations, algebraically isolate, substitutions with units, final statement of prediction Units for Acceleration. In the previous example, you found units of m/s2 for the acceleration. It is important to understand that the two units of time in (m/s)/s (m/s2 is shorthand) play different roles. The second in m/s is just part of the unit for velocity (like hour in km/h). The other second is the unit of time we use when describing how much the velocity changes in one unit of time. For problem solving convenience, our new equation for acceleration is often written as: vf = vi + a∆t 3. Explain. Show the algebraic steps that start from the equation a = ∆v/∆t and lead to vf = vi + a∆t. 4. Solve. You are driving west along the 401 and want to pass a large truck. You floor the gas pedal and begin to speed up. You start at 102 km/h, accelerate at a steady rate of 2.9 (km/h)/s (obviously not a sports car). What is your velocity after 5.3 seconds when you finally pass the truck? 56  A: Pictorial Representation B: Physics Representation Sketch showing events, describe events, coordinate system, label givens & Motion diagram, velocity graph, velocity vectors, event numbers unknowns using symbols, conversions C: Word Representation Describe motion (no numbers), assumptions, estimated result (no calculations) D: Mathematical Representation Describe purpose of math, complete equations, algebraically isolate, substitutions with units, final statement of prediction Vertical Position. To describe motion in the vertical direction, use the symbol y for the vertical position. All other symbols remain the same. In physics, the symbol x will only be used for horizontal position. The sketch for the situation should show the vertical motion and the coordinate system should show which vertical direction is the +y-direction. The motion diagram and the velocity vectors should point vertically. 5. Solve. A rocket is travelling upwards at a velocity of 413 km/h. Its engine cuts off after which it continues moving up but slows down at a rate of -13.7 m/s2. The scientists launching it want it to reach its highest point (with a velocity of zero) within 30 seconds. Is it able to do so? (Hint: convert v1 in part A) 57  A: Pictorial Representation B: Physics Representation Sketch showing events, coordinate system, label givens & unknowns using Motion diagram, velocity graph, velocity vectors, events symbols, conversions, describe events C: Word Representation Describe motion (no numbers), assumptions, estimated result (no calculations) D: Mathematical Representation Describe purpose of math, complete equations, algebraically isolate, substitutions with units, final statement of prediction 58 Homework: Speeding Up and Slowing Down Name: 1. Interpret. A person walks back and forth in front of a motion detector producing the velocity graph shown below. Six events have been labelled on the graph. For each interval of time, complete each row of the chart. Two of the rows in the chart are explained below: State of motion: choose between rest, constant velocity, or acceleration. SU / SD? If it is accelerating, decide whether it is speeding up (SU) or slowing down (SD). v t ∙1 ∙ 4 6 ∙ 2 ∙ 3 ∙ 5 ∙ Sign of Velocity Sign of Acceleration State of Motion SU / SD? Velocity Graph Position Graph x t ∙1 2 ∙ Position Graph negative Slope: sign Position Graph more shallow Slope: value 59  Position Graph curve shape 2. Represent. Sketch the position graph that corresponds to the velocity graph above. To do this, complete the column under the position graph before sketching that section. Follow the steps below: (a) Position Graph Slope: Sign – is the sign of the position graph slope positive (velocity positive) or negative (velocity negative)? (b) Position Graph Slope: value – is the slope become steeper (speeding up) or more shallow (slowing down)? (c) Position Graph Shape – is it a straight line (constant velocity) or curve (acceleration)? (d) Draw event positions – for your interval, decide whether the person is moving in the positive or negative direction. Draw dots for this interval showing these events and label them with event numbers. (e) Sketch sample slopes – at the start and end of your interval, sketch two sample slopes using the descriptions in that column. (f) Sketch position graph – connect the two sample slopes using the graph shape you decided on. 3. Interpret. Circle each event on the velocity graph where the acceleration changes. Highlight each interval during which the acceleration is constant. Motion 12: Speeding Up or Slowing Down? There is one mystery concerning acceleration remaining to be solved. Our definition of acceleration, Δv/Δt, allows the result to be either positive or negative, but what does that mean? Today we will get to the bottom of this! A: Acceleration in Graphs Your teacher has set-up a cart with a fan on a dynamics track and a motion detector to help create position-time and velocity-time graphs. Let’s begin with a position graph before we observe the motion. The cart is initially moving forward. The fan is on and gives the cart a steady, gentle push which causes the cart to accelerate. Time Position 1. Reminder. What does the slope of a tangent to any position-time graph represent? 2. Interpret. Is the cart speeding up or slowing down? Use the two tangents to the graph to help explain. Tangent Trick for Position Graphs. To help interpret position graphs, we will use the tangent trick. Use a ruler or pencil as the tangent line to a position graph. Interpret the slope of the tangent. Then move the tangent to a new spot along the graph and interpret. Decide if the object is speeding up or slowing down. This trick can also be used decide how to sketch a velocity graph if you have a position graph. 3. Reason. Is the change in velocity positive or negative? What does this tell us about the acceleration? Time Velocity Time Velocity Isaac’s graph Albert’s graph 60 4. Reason. Two students draw a velocity graph based on the position graph above. Which graph do you think best matches the position graph? Explain. 5. Test and interpret. (as a class) Observe the velocity-time graph produced by the computer for this situation. Interpret the motion shown in the velocity graph. In all the following examples, east is the positive direction. Type of graph Feature Value Meaning Sign of velocity values Size of velocity values Time Shape of graph - Velocity + Sign of slope of graph 6. Observe, Predict and Interpret. (as a class) Your teacher will lead you through four different situations involving the cart. You will make observations, make prediction and interpret the results using the chart on the next page. Situation 1 2 3 4 The cart is released from The cart is released from The cart is moving away The cart is moving rest near the motion rest far from the from the detector. The towards the detector. Description detector. The fan exerts a detector. The fan exerts a fan exerts a force on the The fan exerts a force on force on the cart pointing force on the cart cart towards the the cart away from the away from the detector. towards the detector. detector. detector. Sketch with Force F positive negative F F negative positive F Observe Observe Observe Predict Position graph Observe Predict Predict Predict Velocity graph 61  Observe Predict Predict Predict Acceleration graph Slowing down or speeding up? Sign of Velocity Sign of Acceleration 7. Reason. Emmy says, “We can see from these results that when the acceleration is positive, the object always speeds up.” Do you agree with Emmy? Marie says, “No. There’s more to it than that.” Who do you agree with? Explain. 8. Reason. What conditions for the acceleration and velocity must be true for an object to be speeding up? To be slowing down? 9. Reason. The sign of the acceleration always matches the sign of what other quantity in our chart? Always compare the magnitudes of the velocities (the speeds) using the terms faster or slower. Describe the motion of accelerating objects as speeding up or slowing down and state whether it is moving in the positive of negative direction. Other ways of describing velocity often lead to ambiguity and trouble! Never use the d-word (deceleration) - yikes! Note that we will always assume the acceleration is uniform (constant) unless there is a good reason to believe otherwise. 10. Reason. In situation #4, why might it be confusing to interpret the velocity graph by saying, “the velocity is increasing”? What might be better to say? 62 Motion 14: Area and Displacements A graph is more than just a line or a curve. We will discover a very handy new property of graphs which has been right under our noses (and graphs) all this time! A: Looking Under the Graph A car drives south along a straight road at 20 m/s. After 5 s the car passes a streetlight and at 20 s the car passes a bus stop. 1. Describe. Based on what you have learned so far in physics, how can we calculate the displacement of the car between the streetlight and the bus stop? 2. Reason. Suppose instead that the car’s starting velocity was 20 m/s and at 5 s the car began to speed up. In the same 15-s time interval, would the car’s displacement be larger or smaller? Explain. 3. Sketch. Now we will think about this calculation in a new way. Draw and shade a rectangle on the graph that fills in the area between the line of the graph and the time axis, for the time interval of 5 to 20 seconds. 4. Describe. In math class, how would you calculate the area of the rectangle? 5. Interpret. Calculate the area of the rectangle. Note that the length and width have a meaning in physics, so the final result is not a physical area. Use the proper physics units that correspond to the height and the width of the rectangle. What physics quantity does the final result represent? Area under a velocity graph. The area under a velocity-time graph for an interval of motion gives the displacement during that interval. Both velocity and displacement are vector quantities and can be positive or negative depending on their directions. According to our usual sign convention, areas above the time axis are positive and areas below the time axis are negative. B: Applying Our New Tool Our new tool for finding displacements will help us find the answer to a sticky problem: how can we find the displacement of an accelerating object? 63 Consider the graph on the next page that shows the velocity of an object that is speeding up. We want to use this graph to find the displacement of the object between the times ti and tf. The area under this graph has an unusual shape, but we can split up the area into two simpler shapes. 1. Represent. How do we find the area of any rectangle? Write an expression for the area the way you would write it in math class. How can we find the area of the rectangle under this graph? Write a new expression for the area using physics symbols from this graph. Velocity Δt Δv ti tf vf vi Time math class expression: area = physics graph expression: area = 2. Represent. Write an expression for the area of the triangle in two different ways. math class expression: area = physics graph expression: area = Our goal is to create an equation that lets us find the displacement of the object if we know its acceleration. To do that, we need to do a math trick. 3. Represent. Remember our definition of acceleration: a = ∆v/∆t. If we rearrange it, we have: Δv = aΔt. Make a mathematical substitution for ∆v in your physics graph expression for the area of the triangle. Do a bit of algebra work to simplify your expression. area = 4. Represent. Create one expression that describes the total area underneath the graph. area = 5. Interpret. We understand that the area under the graph between those moments in time represents the displacement of the object. Write a final version of your equation. Replace the word “area” with the appropriate physics symbol. ** call your teacher over to check your equation ** The BIG 5 Equations for Constant Acceleration. The equation you have just constructed is one of the five equations for constant acceleration (affectionately known as the BIG five). Together they help relate different combinations of the five variables of motion: Δx, a, vi, vf and Δt. You have encountered one other BIG five so far, (in a disguised form) the definition of acceleration: a = Δv/Δt. Recall that this equation was also constructed by analyzing a graph showing changing velocity! Awesome! 64 6. Evaluate. Would the new equation produce a result that agrees with your response for question A#2? (Suppose the acceleration was +1 m/s2) Explain. Displacement Problems! Use the full solution format to solve these problems. Hint: when choosing an equation (you have a choice of two), think about which quantities you know and which you are trying to find out. 1. Taking Off. A jumbo jet takes flight while travelling down a 1.80 km runway. It barely makes it off the ground after it reaches the end of the runway, taking 37.9 s of time. What is the acceleration of this jet? Give your answer in m/s2 2. Stopping a Muon. A muon (a subatomic particle) moving in a straight line enters a detector with a speed of 5 x 106 m/s and then it slowed down at the rate of 1.25 x 1014 m/s2 in 4 x 10-8 s. How far does it travel while slowing down? (Hint: to slow down, one of your vector quantities will need to be negative. Which one?) Motion 15: The BIG Five Last class we found three equations to help represent motion with constant acceleration. A bit more work along those lines would allow us to find two more equations which give us a complete set of equations for the five motion quantities. A: The BIG Five – Revealed! Here are the BIG five equations for uniformly accelerating motion (the acceleration is constant). The BIG Five vi vf Δx a Δt vf = vi + aΔt Δx = viΔt + ½aΔt2 Δx = vfΔt - ½aΔt2 Δx = ½(vi + vf)Δt vf 2 = vi2 + 2aΔx 1. Observe. Fill in the chart with √ and × indicating whether or not a motion quantity is found in that equation. 2. Find a Pattern. How many quantities are related in each equation? 3. Reason. If you wanted to use the first equation to calculate the acceleration, how many other quantities would you need to know? 4. Describe. Define carefully each of the kinematic quantities in the chart below. vi vf Δx a Δt 65 5. Reason. What condition must hold true (mentioned in the previous investigation) for these equations to give reasonable or realistic results? B: As Easy as 3-4-5 Solving a problem involving uniformly accelerated motion is as easy as 3-4-5. As soon as you know three quantities, you can always find a fourth using a BIG five! Write your solutions carefully using our solution process. Use the chart to help you choose a BIG five. Here are some sample problems that we will use the BIG five to help solve. Note that we are focusing on certain steps in our work here – in your homework, make sure you complete all the steps! 1. Solve. Your teacher has an inclined track set up at the front of the room. Your teacher will release a cart from rest at the top of the track. Your group must choose a position along the track. Label this position with a sticky-note that includes your group number and the displacement of the cart when it reaches that position. Your challenge is to predict the cart’s speed at that position. Your teacher will give you the cart’s acceleration. When you are finished, add your prediction to your sticky-note. A: Pictorial Representation Sketch showing events, describe events, coordinate system, label givens & unknowns with symbols, conversions B: Physics Representation x v t t a t Motion diagram, motion graphs, velocity vectors, event numbers D: Mathematical Representation Describe purpose of math, complete equations, algebraically isolate, substitutions with units, final statement of prediction 66 C: You’ve Got Problems: complete these problems on a separate solution sheet (1) Crash Test. An automobile safety laboratory performs crash tests of vehicles to ensure their safety in high-speed collisions. The engineers set up a head-on crash test for a dummy buckled into a Smart Car that collides with a solid barrier. The engineers know the dummy initially travels south at 100 km/h and moves 0.78 m south during the collision. The engineers have a couple of questions: How much time does the collision take? What is the dummy’s acceleration during the collision? Hint: Event 1 = dummy begins to slow down, Event 2 = Dummy stops (2) Microscopic physics. All cell biology works according to the laws of physics! A sodium ion (3.817x10-26 kg) arrives near an opening in a cell. You may assume it is initially at rest. Electric forces cause it to speed up and travel towards the cell opening. As a result, it travels 1.48x10-7 m in 0.512 s. What is the acceleration of the sodium ion? (3) The Track. A cart is placed at the bottom of an inclined track. It uses a spring to launch itself up the incline with a speed of 0.79 m/s. While travelling up and down the incline, the cart has an acceleration of 0.54 m/s2. How much time does it take to make the complete trip up and back down to its starting position? (Hint: this is a one-step problem) 67