Motion in Two Dimensions PDF

Unit 4: Motion in Two Dimensions Learning Objectives DepEd Competency In this lesson, you should be able to do the Describe motion using the concept of relative velocities in 1D...

Unit 4: Motion in Two Dimensions Learning Objectives DepEd Competency In this lesson, you should be able to do the Describe motion using the concept of relative velocities in 1D and 2D following: (STEM_GP12KIN-Ic-20). Explain what relative velocity is. Solve the relative velocity of objects in one dimension. Solve the relative velocity of objects in two dimensions. Warm Up Relative Velocity Race 5 minutes The simulation you are about to interact with demonstrates relative velocity in one dimension. Two women, Mia and Brandi are competing in an “out-and-back race”. Mia, designated by the red dot in the simulation, runs along a moving sidewalk, whereas Brandi, represented by the blue dot, moves across a non-moving floor. Materials relative velocity race simulation pen and paper 4.1. Motion Descriptors in Two Dimensions 2 Unit 4: Motion in Two Dimensions Procedure 1. Refer to the simulation. A screenshot, showing its initial conditions, is presented below: Relative Velocity in 1-Dimension Andrew Duffy, “Relative Velocity in 1-Dimension,” http://physics.bu.edu/~duffy/HTML5/relative_velocity_race.ht ml http://www.onlinegamelinkhere.com, last accessed on March 01, 2020 Fig. 4.1.1. Relative velocity race simulation 2. Set the speed of the moving sidewalk to 2 m/s. 3. Next, set Mia’s speed on the moving sidewalk to 5 m/s. 4. Then, set Brandi’s running speed on the nonmoving floor to 5 m/s. 5. Finally, click “Play”. 4.1. Motion Descriptors in Two Dimensions 3 Unit 4: Motion in Two Dimensions Guide Questions 1. Who do you think won the race and why? 2. What should the losing runner have done to win the race? 3. How do you think the moving sidewalk changed Mia’s perception of Brandi’s motion? Learn about It! When is velocity considered to be relative? Relative velocity ( ) refers to the velocity of an object A from the frame of reference of another observer or object B. The general formula for relative velocity is: Equation 4.1.1 This equation also tells us that the resultant velocity is the summation of all the vectors under consideration. The object reference is assigned a negative value when it is moving in the opposite direction of the object observed. In such case, the relative velocity is expressed as follows: Relative Velocity in One Dimension Relative velocity in one dimension entails the individual motions of an object and its observer in a linear direction, i.e., across a straight line whose motions have only three possible directions: in the same direction, towards each other, and opposite each other. 4.1. Motion Descriptors in Two Dimensions 4 Unit 4: Motion in Two Dimensions Position of the Point Object Visualizing the motions of the two objects in consideration and distinguishing how to arrange the subscripts in the relative velocity variable ( or ) are extremely crucial in identifying the relative velocity of any given object. The first subscript must represent the object possessing the velocity. It is the one whose motion is being observed. On the other hand, the second subscript represents the object or observer with respect to which the difference in velocities is being calculated. It is also called the frame of reference. Thus, the meaning of the subscripted velocities may vary depending on the position from where the motion is viewed and must thus be expressed accordingly, as follows: = velocity of object A relative to object B; = velocity of object B relative to object B, and = velocity of object A relative to object C. Solving Relative Velocity in 1D Consider, for instance, that you are seated inside a train with a velocity of 15 m/s in the eastward direction. 15 m/s east is the velocity of the train with respect to the Earth’s surface. If you will walk towards the east with a velocity of 12 m/s, then your velocity relative to the train changes, as shown below: (where A is your velocity, and B is the train’s) 𝑚 𝑣𝐴𝐵 = 27 𝑠 4.1. Motion Descriptors in Two Dimensions 5 Unit 4: Motion in Two Dimensions If another train, this time moving in the westward direction, arrives with the velocity of 16 m/s, the velocity of the first train from the perspective of the second train changes, as shown below: (where B is the velocity of the first train, and C is the velocity of the second train) The frame of reference assumes a negative value, since it is moving in the opposite direction. The velocity of the first train is -1 m/s with respect to the second train moving west. Tips Note that the negative value in the reference object signifies that “it has moved away from the object observed at a given velocity in the opposite direction”. 4.1. Motion Descriptors in Two Dimensions 6 Unit 4: Motion in Two Dimensions Remember Velocities vary depending on the frames of reference from where motion is observed. However, the physical situation remains the same. This is where the word “relative” comes into play. When relative velocity is under consideration, the velocity of the reference frame must always be constant and not accelerating, i.e., it must either be zero (at rest) or have a uniform speed in a straight line. What does relative velocity in one dimension mean? Relative Velocity in Two Dimensions On occasion, objects under consideration change their direction, such as in the case of airplanes encountering side winds, or in motorboats being swayed by the river’s current. In these circumstances, the relative velocity of objects is still determined by the equation , but the magnitude of the resultant velocity is calculated using the Pythagorean theorem: Equation 4.1.2 Consider an airplane traveling south with a velocity of 113 km/hr. It experiences a side wind that has a speed of 22 km/hr. To identify the resultant velocity, you must get the vector sum of the two velocities under consideration. This was the same process used for entities in one dimension. However, since the two vectors are at right angles to each other, the Pythagorean Theorem can be used to determine the magnitude of the resultant velocity, as follows: 4.1. Motion Descriptors in Two Dimensions 7 Unit 4: Motion in Two Dimensions Next, to calculate the direction of the relative velocity, use the tangent function, as follows: Therefore, the resultant velocity of the plane relative to the observer on the ground is 115.12 km/hr at 11 degrees. How do we compute the relative velocity of an object moving perpendicularly from or to its frame of reference? 4.1. Motion Descriptors in Two Dimensions 8 Unit 4: Motion in Two Dimensions Let’s Practice! Example 1 A box of pencils is put on a conveyor belt that runs with a velocity of 0.059 mph. If you set a box of marbles to run on the same conveyor belt, what is the velocity of the box of pencils relative to the box of marbles? Solution Step 1: Identify what is required in the problem. You are asked to calculate the relative velocity (vAB) of the box of pencils (A) with respect to the velocity of the box of marbles (B). Step 2: Identify the given in the problem. Even if the conveyor belt is moving at a speed of 0.059 mph, the velocity of the box of pencils relative to its fixed place in the conveyor belt is still 0 mph. The same is true for the box of marbles, thus: Step 3: Write the working equation. Step 4: Substitute the given values. Step 5: Find the answer. Hence, the velocity of the box of pencils relative to the velocity of the box of marbles is zero, which concludes that both of these boxes are moving neither away or toward each other. 4.1. Motion Descriptors in Two Dimensions 9 Unit 4: Motion in Two Dimensions 1 Try It! George rode an escalator with a velocity of 0.06 mph. If Anna will ride the same escalator, what will be her velocity with respect to George’s? Example 2 A yellow cab driving along the highway with a velocity of 29 mph overtakes another car. From a passenger’s perspective, the relative velocity of the second car relative to the yellow cab is 4 mph. Identify the second car’s velocity. Solution Step 1: Identify what is required in the problem. You are asked to calculate the velocity of the second car. Step 2: Identify the given in the problem. The velocity of the yellow cab (VA) and the relative velocity of the second car (VB) with respect to the yellow cab are given. Step 3: Write the working equation. Step 4: Substitute the given values. Note that the second car assumes a negative velocity, because, as the yellow cab moves past it, it would appear to be moving backwards from the vantage of that car. Step 5: Find the answer. 4.1. Motion Descriptors in Two Dimensions 10 Unit 4: Motion in Two Dimensions Hence, the velocity of the second car is –25 mph. This also concludes that the second car is 4 mph slower than the yellow cab. 2 Try It! A cabin cruiser (A) whose speedometer reads 23 mph runs off across the river, while a man tries to run after it from the riverbank with a velocity of 13 mph. Determine the velocity of the cabin cruiser relative to the first man. Example 3 A motorboat heads north with a velocity of 5 m/s on a river whose current is moving at a velocity of 2 m/s to the east. What is the velocity of the motorboat relative to a person watching it from the riverbank? Solution Step 1: Identify what is required in the problem. You are asked to calculate the relative velocity (vAB) of the motorboat (A) affected by the current (B) with respect to the person watching it from the riverbank (C). Step 2: Identify the given in the problem. First, identify the resultant velocity of the motorboat (A) and the current (B): Convert the following velocities to mph. Step 3: Write the working equation. 4.1. Motion Descriptors in Two Dimensions 11 Unit 4: Motion in Two Dimensions Step 4: Substitute the given values. Step 5: Find the answer. Calculate the direction of the resultant, as follows: Thus, the resultant velocity of the motorboat relative to the observer on the bank is 12.06 mph at 68.2º. 3 Try It! A motorcycle is heading north with a velocity of 87 km/ hr towards an intersection. A bus is travelling west toward the same intersection at a velocity of 92 km/ hr. What is the velocity of the bus relative to the motorcycle? Key Points ___________________________________________________________________________________________ Relative velocity (vBA) refers to the velocity of an object A from the frame of reference of another observer or object B. When relative velocity is under consideration, the velocity of the reference frame must always be constant and not accelerating, i.e., it must either be zero (at rest) or have a uniform speed in a straight line. A reference object that is moving away from the object observed assumes a negative velocity. 4.1. Motion Descriptors in Two Dimensions 12 Unit 4: Motion in Two Dimensions To calculate for the magnitude of the relative velocity of objects moving in two dimensions, use the Pythagorean theorem. To calculate the direction of the resultant velocity, the tangent function is used. ___________________________________________________________________________________________ Key Formulas ___________________________________________________________________________________________ Concept Formula Description Relative Velocity in Use this formula to solve for One Dimension where the relative velocity of vAB is the relative velocity of Object A with respect to the A compared to B; reference, Object B when vA is the velocity of the Objects A and B are object observed, and moving in a linear course. vB is the velocity of the frame of reference. Relative Velocity in Use the same general equation Use this formula to solve for Two Dimensions for vAB, but compute the the relative velocity of resultant velocity using the Object A with respect to the Pythagorean theorem. reference, Object B when Objects A and B are moving in two dimensions, where e.g., perpendicularly. vAB is the relative velocity of A compared to B; vA is the velocity of the object observed, and vB is the velocity of the 4.1. Motion Descriptors in Two Dimensions 13 Unit 4: Motion in Two Dimensions frame of reference. To identify the direction: ___________________________________________________________________________________________ Check Your Understanding A. Identify whether each statement is true or false. 1. Relative velocity may or may not have a negative value. 2. In the variable vAB, A refers to the frame of reference. 3. An object travelling to the opposite direction of another object assumes a positive value in the equation of relative velocity. 4. The direction of the resultant velocity is the clockwise angle of rotation that the resultant vector makes, due west. 5. People are bound to observe the same velocity regardless if their frames of reference are the same or not. 6. People are bound to observe the same velocity regardless if they are moving or not. 7. In the variable vBA, B refers to the object in motion being observed. 8. Relative velocity is important in determining the differences between the velocities of moving objects. 9. Relative velocity is necessary to identify whether motion has transpired in a given setting or not. 10. An observer moving in a similar direction with another moving object assumes a positive value in the relative velocity equation. 4.1. Motion Descriptors in Two Dimensions 14 Unit 4: Motion in Two Dimensions 11. An object travelling to the opposite direction of another object will always have a 90º in the equation of relative velocity. 12. To determine the angle between object A and object B, the function. 13. The horizontal and vertical components of relative velocity are independent. 14. Relative velocity is applicable to everything that moves. 15. Relative velocity is dependent on time. B. Solve the following problems. 1. A motorcycle speeding at 135 km/h overtakes a taxi running with a velocity of 97 km/h. What is the velocity of the motorcycle from the perspective of the taxi’s driver? 2. A cargo plane heading east flies with a velocity of 785 mph. A private plane proceeds west with a velocity of 815 mph. Calculate the relative velocity of the private plane from the point of view of the cargo plane. C. Contrast the different velocities a driver of a moving bus could have at one particular moment relative to various frames of reference. D. Using your knowledge of relative velocity, formulate at least 1 precautionary guideline for road safety to avoid accidents such as collisions. Challenge Yourself Answer the following questions. 1. Using the notion of relative velocity, argue why when you are seated inside a moving vehicle, objects outside you seem to be moving in a backwards direction. 2. Is there such a thing as motion at rest? Why or why not? 4.1. Motion Descriptors in Two Dimensions 15 Unit 4: Motion in Two Dimensions 3. How important is relative velocity in road safety? 4. How important is relative velocity in air traffic control? 5. How would you relate the idea of varying frames of reference to the decisions people arrive at in life? Bibliography Hewitt, Paul G. 2010. Conceptual Physics (11th ed). New York: Pearson Education. Holt, R., Serway, R., & Faugn, J. (2006). Physics. Austin, TX. Macalalad, E. P. and Vergara, R. L. 2011. Exploring the Realms of Science: Physics. Valenzuela City: JO-ES Publishing House, Inc. Vincent P. Coletta. Physics Fundamentals. Physics Curriculum & Instruction, Inc.: 2010. Walker, J., Halliday, D., & Resnick, R. (2011). Fundamentals of physics. Hoboken, NJ: Wiley. Key to Try It! 1. 0 mph 2. 10mph 3. 64.0 mph at 62.9 degrees 4.1. Motion Descriptors in Two Dimensions 16