Module 4 - Physics I PDF

Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 17 Subject Teacher: Julieta A. Malones Module IV: Kinematics Module Outline 4.1 Motion 4.2 Distance and Displacement 4.3 Speed and Velocity 4.4 Acceleration 4.5 Uniformly Accelerated Motion 4.6 F...

Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 17 Subject Teacher: Julieta A. Malones Module IV: Kinematics Module Outline 4.1 Motion 4.2 Distance and Displacement 4.3 Speed and Velocity 4.4 Acceleration 4.5 Uniformly Accelerated Motion 4.6 Free Falling Objects Intended Learning Outcomes Upon the completion of this module, the students should be able to:  Identify concepts related to motion  Distinguish between distance and displacement and speed and velocity  Observe and describe the motion of objects  Describe straight-line motion in terms of velocity and acceleration  Solve problems involving straight-line motion with constant acceleration, including free fall problems  Explain the motion of falling bodies without air friction  Calculate the velocity of an object after it has fallen freely for a known interval of time Introduction Studies of Motion In Classical Mechanics, there are two fields of studying motion: Kinematics and Dynamics. Given Newton's laws of motion, it is understandable that these two fields would be divided by the main effect on motion and velocity: force. This module focuses on kinematics – the study that deals with the description of motion. Kinematics uses the following basic concepts of motion: distance, displacement, speed, velocity and acceleration. As mentioned, kinematics describes the position and motion of an object as a function of time, but does not include the causes or factors that affect the motion. The two methods by which the motion of an object can be described are those using mathematical equations and graphical analysis. Learning Content 4.1 Motion Source: https://www.tutorialspoint.com/physics_part1/images/motion.jpg Everything moves. Even things that appear to be at rest move. Motion is the change in an object's position over a specific period of time. Motion, normally, is described in terms of displacement, velocity, acceleration, distance, time, and speed. For us to adequately describe motion, we must be able to check where the body is located within a given frame of reference. A reference frame is a physical entity such as the earth's surface, the deck of a ship or a moving vehicle, to which the position and motion of an object is relative.  Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 18 The idea that a description of motion depends on the reference frame of the observer has been known for hundreds of years. The 17th-century astronomer Galileo Galilei was one of the first scientists to explore this idea. Galileo suggested the following thought experiment: Imagine a windowless ship moving at a constant speed and direction along a perfectly calm sea. Is there a way that a person inside the ship can determine whether the ship is moving? You can extend this thought experiment by also imagining a person standing on the shore. How can a person on the shore determine whether the ship is moving? Galileo came to an amazing conclusion. Only by looking at each other can a person in the ship or a person on shore describe the motion of one relative to the other. In addition, their descriptions of motion would be identical. A person inside the ship would describe the person on the land as moving past the ship. The person on shore would describe the ship and the person inside it as moving past. Galileo realized that observers moving at a constant speed and direction relative to each other describe motion in the same way. Galileo had discovered that a description of motion is only meaningful if you specify a reference frame. Grasp Check Imagine standing on a platform watching a train pass by. According to Galileo’s conclusions, how would your description of motion and the description of motion by a person riding on the train compare? a. I would see the train as moving past me, and a person on the train would see me as stationary. b. I would see the train as moving past me, and a person on the train would see me as moving past the train. c. I would see the train as stationary, and a person on the train would see me as moving past the train. d. I would see the train as stationary, and a person on the train would also see me as stationary. 4.2 Distance and Displacement Motion may be described by specifying how far something has traveled in changing position and time. The total path length traversed by an object moving from one location to another is known as distance while the separation of that object and a reference point is known as displacement. Distance is a scalar quantity which has only magnitude while displacement is a vector quantity which has both magnitude and direction. We often want to be more precise when we talk about position. The description of an object’s motion often includes more than just the distance it moves. For instance, if it is a five kilometer drive to school, the distance traveled is 5 kilometers. After dropping you off at school and driving back home, your parent will have traveled a total distance of 10 kilometers. The car and your parent will end up in the same starting position in space. The net change in position of an object is its displacement, or ∆d. The Greek letter delta, ∆ , means change in. Illustration source: Physics (2020) If you are describing only your drive to school, then the distance traveled and the displacement are the same—5 kilometers. When you are describing the entire round trip, distance and displacement are different. When you describe distance, you only include the magnitude, the size  Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 19 or amount, of the distance traveled. However, when you describe the displacement, you take into account both the magnitude of the change in position and the direction of movement. Displacement Problems Hopefully you now understand the conceptual difference between distance and displacement. Understanding concepts is half the battle in physics. The other half is math. A stumbling block to new physics students is trying to wade through the math of physics while also trying to understand the associated concepts. This struggle may lead to misconceptions and answers that make no sense. Once the concept is mastered, the math is far less confusing. So let’s review and see if we can make sense of displacement in terms of numbers and equations. You can calculate an object's displacement by subtracting its original position, d0, from its final position df. In math terms that means ∆d = df – d0 If the final position is the same as the initial position, then ∆d = 0 Self-Check Calculating Distance and Displacement A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is her displacement? (b) What distance does she ride? (c) What is the magnitude of her displacement? 4.3 Speed and Velocity Speed There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner’s speed?” cannot be answered without an understanding of other concepts. In this section we will look at time, speed, and velocity to expand our understanding of motion. A description of how fast or slow an object moves is its speed. Speed is the rate at which an object changes its location. Like distance, speed is a scalar because it has a magnitude but not a direction. Because speed is a rate, it depends on the time interval of motion. You can calculate the elapsed time or the change in time, ∆t , of motion as the difference between the ending time and the beginning time. ∆t = tf – t0 The SI unit of time is the second (s), and the SI unit of speed is meters per second (m/s), but sometimes kilometers per hour (km/h), miles per hour (mph) or other units of speed are used. When you describe an object's speed, you often describe the average over a time period. Average speed, vavg, is the distance traveled divided by the time during which the motion occurs. Vavg = You can, of course, rearrange the equation to solve for either distance or time time = distance = x time (s)  Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 20 150 km in 3.2 hrs Suppose, for example, a car travels 150 kilometers in 3.2 hours. Its average speed for the trip is Vavg = =. =46.875 km h ≈ 47 km h A car's speed would likely increase and decrease many times over a 3.2 hour trip. Its speed at a specific instant in time, however, is its instantaneous speed. A car's speedometer describes its instantaneous speed. Velocity The vector version of speed is velocity. Velocity describes the speed and direction of an object. As with speed, it is useful to describe either the average velocity over a time period or the velocity at a specific moment. Average velocity is distance traveled divided by the total time elapsed in traveling that distance. ∆ v̅ = =∆ = Velocity, like speed, has SI units of meters per second (m/s). The difference between speed and velocity is that speed is a scalar quantity whereas velocity is a vector which means it has a direction. Sample Problem Rachel watches a thunderstorm from her window. She sees the flash of lightning bolt and begins counting the seconds until she hears the clap of thunder 5.0 seconds later. Assume that the speed of sound in air is 340.0 m/s and the light was seen instantaneously. How far away was the lightning bolt? Given: vs = 340.0 m/s t = 5.0 s Find: distance Solution: distance = v x time d = (340.0 m/s) (5.0 s) d = 1700 m Recall that instantaneous speed is the speed at a specific time. A moving object’s instantaneous speed is always equal to the magnitude of its instantaneous velocity. An airplane whose instantaneous speed is 640 km/h might have an instantaneous velocity of 640 km/h to the west or 640 km/h to the north, or it might be flying in a circle at 640 km/h, but the magnitude of the airplane’s instantaneous velocity is the same in each case. Grasp Check Which of the following fully describes a vector and a scalar quantity and correctly provides an example of each? a. A scalar quantity is fully described by its magnitude, while a vector needs both magnitude and direction to fully describe it. Displacement is an example of a scalar quantity and time is an example of a vector quantity. b. A scalar quantity is fully described by its magnitude, while a vector needs both magnitude and direction to fully describe it. Time is an example of a scalar quantity and displacement is an example of a vector quantity. c. A scalar quantity is fully described by its magnitude and direction, while a vector needs only magnitude to fully describe it. Displacement is an example of a scalar quantity and time is an example of a vector quantity. d. A scalar quantity is fully described by its magnitude and direction, while a vector needs only magnitude to fully describe it. Time is an example of a scalar quantity and displacement is an example of a vector quantity  Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 21 4.4 Acceleration For the motion of an object in which the velocity changes in either magnitude (speed) or direction or both, you will have a new quantity known as acceleration which is a vector quantity. An object is accelerating when it speeds up, slows down (usually called deceleration) or changes direction. Just as velocity is the rate of change of displacement and time, acceleration is the rate of change of velocity with time. Consider this case. A car is at rest. The driver starts the engine. The car starts slow and after a second it is moving at 5 km h. After a second, the car’s speedometer indicates the car’s speed is 15 km h. Another second, the car’s speed is 30 km h, then 50 km h. The changing motion of the car can be described in terms of acceleration. We define acceleration as the rate of change of speed. In equation, Where vf is the final velocity and vi is the initial acceleration = velocity. a= The unit for acceleration is m/s2 ∆ In our example, the car increased by 5 km/h after one second, then 15 km/h, and finally by 20 km/h in the third second. You can see, it did not accelerate at the same rate. Thus it is appropriate to use the term average acceleration. Hence, ̅= ∆ Applying this equation in the above example, the car’s average acceleration is a̅ = ∆ = 12.5 or 3.5 m/s2 Sample Problem Michael is driving his sports car at 30 m/s when he sees a dog on the road ahead. He slams on the brakes and comes to a stop in 3.0 seconds. What was the acceleration of Michael's car? Given: vi = 30 m/s vf = 0 ∆t = 3.0 s Find: acceleration (a) Solution: a= =. a = -10 m/s2 The car slows down at the rate of 10 m/s every second. This is negative acceleration which is known as deceleration. 4.5 Uniformly Accelerated Motion Linear motion is the most straightforward kind of one-dimensional motion. As Newton’s first law of motion suggests, an object will either be at rest or continue to move in a straight line with a uniform velocity unless and until an external force is applied to it. When a body moves in a plane or on a straight line, three parameters are used to describe its motion – distance, velocity, and acceleration. Distance or displacement is self-explanatory. Velocity represents the rate of change of position, while acceleration represents the rate of change of velocity. All three quantities are vector quantities. Acceleration can be uniform or non- uniform. A uniform acceleration has a constant value and direction. It is essential to know the equation of motions that describe the motion of an object under uniform acceleration.  Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 22 Uniform acceleration is the acceleration that does not vary with time. In such cases, the rate of change of velocity remains constant. Since acceleration is a vector quantity, even the direction of motion remains the same in the case of constant acceleration. Since the body is moving in a single direction with a constant magnitude of acceleration, vector notations can be dropped. Some examples of constant acceleration include 1. Free-falling object. 2. A ball rolling down a frictionless slope. 3. A bicycle whose brakes have been engaged. Equations of Uniformly Accelerated Motion All problems that involve straight-line motion under constant acceleration can be solved by using the formulas developed thus far in this chapter. These formulas are listed in Table 4.1 for convenience. Depending on a problem, a particular formula has to be rewritten to suit a particular combination of known and unknown quantities. Table 4.1 Type of Motion Behavior of Physical Quantities Equation  Increasing or decreasing at displacement d v t 2 Constant Acceleration or  Increasing or decreasing or Uniformly Accelerated magnitude of velocity v v d Motion  Constant speed but changing 2a direction  Constant acceleration a = constant = Sample Problem Albert is riding his scooter at a velocity of 80 km/h, when he sees an old woman crossing the road 45 m away. He immediately steps hard on the brakes to get the maximum deceleration of 7.5 m/s2. How far will he go before stopping? Will he hit the old woman? Given: vi = 80 km/h or 22.22 m/s vf = 0 a = -7.5 m/s2 Find: d Solution: d. =.. = d =32.92 m Since the old woman is 45 m away, Albert will be able to stop without hitting her. Self-Check A car speeds up from 40 km/h to 55 km/h to overtake a truck. If this requires 15 s, what is the (a) acceleration and (b) distance traveled by the car?  Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 23 4.6 Free Falling Objects What goes up must come down. It is a common observation that when you toss a coin up, it will come down after some time. What makes it come down? Which way is up? Which way is down? When we say, "something is falling down," we mean, it is falling toward the ground. Since the earth is round, therefore, falling down means moving toward the center of the earth. The apple falls down because it is pulled by the earth. The apple is moving toward the center of the earth. Source: https://search-static.byjusweb.com/question-images/byjus/infinitestudent- images/ckeditor_assets/pictures/684470/original_37_Apple_Falling_down_28129.jpg In the late 16th century, it was generally believed that heavier objects would fall faster than lighter ones. But Galileo (1564-1642) thought differently about this idea. He hypothesized that two objects would fall at the same rate regardless of their mass. His work is an example of what is known as scientific method. The famous story about Galileo dropping two different objects from the Tower of Pisa and observing them fall and reach the ground at the same time is almost certainly a legend. Given the height of the Tower of Pisa, the two objects will not reach the ground at the same time due to effects of air resistance. In his experiment, Galileo measured the speed of falling object by allowing metal balls to roll down an inclined plane and timing them with a water clock. He was able to observe that, in the absence of air resistance, heavy and light objects will fall at the same time and, in the absence of friction, a moving object will maintain its motion unless acted upon by a retarding force. You can perform an equivalent experiment by using a notebook and a sheet of paper. Hold a notebook and a sheet of paper at equal heights from the floor and drop them simultaneously. Which falls faster? Now, repeat the experiment, but this time, crumple the paper into a ball. How do the results differ? Figure 4.1 Demonstrating the effect of air resistance on an object’s acceleration Illustration source: Practical and Explorational Physics: Modular Approach 2nd Edition (2010) In the first case, the notebook fell faster than the sheet of paper. The second case shows that, in the absence of (or with negligible) air resistance, two objects would fall at the same rate, regardless of their weights.  Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 24 Galileo further deduced from his observations that time of fall increases as height of fall increases. He also realized that his experiment showed how air resistance affects the acceleration of a falling object (on Earth). If there is no air resistance and gravity is the only thing that affects a falling object, such an object is said to be in free fall. Analyzing motion for objects in free fall Free fall is a special case of motion with constant acceleration, because acceleration due to gravity is always constant and downward. This is true even when an object is thrown upward or has zero velocity. For example, when a ball is thrown up in the air, the ball's velocity is initially upward. Since gravity pulls the object toward the earth with a constant acceleration g, the magnitude of velocity decreases as the ball approaches maximum height. Figure 4.2. Direction of velocity and acceleration for a ball thrown up in the air. Acceleration from gravity is always constant and downward, but the direction and magnitude of velocity change. Illustration source: https://cdn.kastatic.org/ka-perseus-images/9965a9610a7572e882a719117cbe160e4efde12d.png At the highest point in its trajectory, the ball has zero velocity (see Figure 4.2) , and the magnitude of velocity increases again as the ball falls back toward the earth. The force of gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called acceleration due to gravity. Acceleration due to gravity is so important that its magnitude is given its own symbol, g. It is constant at any given location on Earth and has the average value g=9.8 m/s2 (or32.2ft/s2). Assuming no air resistance, all problems involving motion of falling objects can be solved by using the equations for accelerated motion. The displacement of falling objects in a given period of time is computed by the equation: d = vi t + The final velocity of falling objects can be calculated by the equations: vf 2 = vi 2 + 2gd vf = vi + gt  Physics I – Mechanics, Waves, Heat, Electricity, and Magnetism 25 Sample problem The time a male bungee jumper is freely falling is 1.5 seconds. a) What is the velocity of the jumper at the end of 1.5 seconds? b) How far does he fall? Given: vi = 0 (jumper starts from rest) t = 1.5 s g = 9.8 m/s2 Find: a) vf b) d Solution: a) vf = vi + gt = 0 + (9.8 m/s2) (1.5 s) vf =14.7 m/s or 15 m/s b) d = vi t + = 0 + (9.8 m/s2) (1.5 s)2 = 0 + (9.8 m/s2) (2.25 s2) d = 11.025 m or 11 m Self-Check A stone is thrown upward with an initial speed of 16 m/s. a) What will its maximum height be? b) When will it return to the ground? - End of Module 4 – References: Padua, A., & Crisostomo, R. (2010). Practical and Explorational Physics: Modular Approach (2nd edition). Vibal Publishing House, Inc. Understanding Motion in Physics. (2020). In ScienceAid. Retrieved Aug 3, 2023, from https://scienceaid.net/Understanding_Motion_in_Physics Tutorials Point. (n.d.). Physics – Motion. Retrieved Aug 3, 2023 from https://www.tutorialspoint.com/physics_part1/physics_motion.htm Urone, P., & Hinrichs, R. (2020). Physics. OpenStax National Institute for Science and Mathematics Education Development. (2004). Physics Textbook. Book Media Press, Inc. BYJU’S. (n.d.). Motion in a Straight Line. Retrieved Aug 2, 2023 from https://byjus.com/physics/motion-in-a-straight- line/#:~:text=If%20a%20body%20travels%20in%20a%20straight%20line,rate%20of% 20change%20of%20its%20velocity%20remains%20constant. Anjalsihukla. (n.d.). Uniformly Accelerated Motion. Geek forGeeks. Retrieved Aug 2, 2023 from https://www.geeksforgeeks.org/uniformly-accelerated-motion/ Khan Academy (n.d.). Freefall review. Retrieved Aug 2, 2023 from https://www.khanacademy.org/science/in-in-class11th-physics/in-in-class11th-physics- motion-in-a-straight-line/in-in-class11-objects-in-freefall/a/freefall-ap1 Beiser, A. (1992). Modern Technical Physics (6th edition). Addison-Wesley Publishing Company, printed by Echanis Press, Inc.