2.9 Motion in Two Dimensions
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1 2 KINEMATICS 2.9 Motion in Two Dimensions Name: 2.9 Motion in Two Dimensions Velocity An object is moving around an oval track. Sketch the trajectory of the object on a large sheet of paper. Make your diagram large! Choose a point to serve as the origin for your coordinate system. Label that point O. Select two locations of the object that are about one eigth of the oval apart and label them A and B. After solving this problem, copy your diagram into this space: 1. Draw the position vectors for the locations A and B, and draw the vector that represents the displacement from A to B. 2. Describe how to use the displacement vector to determine the direction of the average velocity of the object between A and B. Draw a vector to represent the average velocity. 3. Choose a point on the oval between points A and B and label it B. Find the average velocity between A and B. Is it different from the average velocity between A and B? If so, how? 4. What happens as you move point B closer and closer to point A? 5. Describe the direction of the instantaneous velocity of the object at point A. Does your answer depend on whether the object is speeding up, slowing down or moving at constant speed? 63
2 2.9 Motion in Two Dimensions 2 KINEMATICS Acceleration for motion with constant speed Let s say the object from before is moving around the track at constant speed. Draw vectors to represent velocity at two points on the track that are fairly close. Label the two points C and D. 6. Find the difference of the two velocity vectors v C and v D, and call it v. 7. Is the angle formed by the head of v C and the tail of v greater than, less than or equal to 90? 8. As point D is chosen to lier closer and closer to point C, how does the above angle change? 9. How would you find average acceleration from change in velocity v? Draw a vector to represent a avg between C and D. 10. What happens to the average acceleration as you choose point D to be closer and closer to C? What about the angle between a and v C? 64
3 2 KINEMATICS 2.9 Motion in Two Dimensions Moving in a Circle Let s look at the details of moving in a circle at constant speed. 11. Sketch the velocity and acceleration vectors at the three points marked below: B A C 12. The term centripetal means center-seeking, centrifugal means center-fleeing. Does either work to describe the acceleration you sketched above? If so, which one? 65
4 2.9 Motion in Two Dimensions 2 KINEMATICS 13. On the next page are two circles of different radii on which two objects travel with the same speed. One radius is twice the length of the other. In this section, we will see how the radius affects the magnitude of the acceleration. (a) Pick two points on each circle that are separated by the same t. (Separate them by about an 1/8th of a circle on the smaller one.) How can you be sure that the t is the same for both circles? (b) At each of the four points, sketch the velocity vector. Lengths of velocity vectors must have the correct relative length. What length are you choosing for each velocity vector? (The drawings will be clearer if your vectors aren t too small, say 1 to 2 inches.) v 1,small = v 1,big = v 2,small = v 2,big = (c) Use those velocity vectors to calculate the average acceleration for each circle. How big is each acceleration (in units of your sketch, ie., inches or centimeters)? a small = a big = (d) State in words why the radius must affect the magnitude of the acceleration. (e) Is what we have seen here consistent with what you learned about the effect of curvature on acceleration in the first part of this activity? 66
5 2 KINEMATICS 2.9 Motion in Two Dimensions 67
6 2.9 Motion in Two Dimensions 2 KINEMATICS 14. On the next page are two circles of the same radius on which two particles travel with different speeds. One speed is twice the magnitude of the other. In this section, we will see how the speed affects the magnitude of the acceleration. (a) Pick two points on each circle that are separated by the same t. (Separate them by about an 1/8th of a circle on the faster one.) How can you be sure that the t is the same for both circles? (b) At each of the four points, sketch the velocity vector. Lengths of velocity vectors must have the correct relative length. What length are you choosing for each velocity vector? (The drawings will be clearer if your vectors aren t too small, say 1 to 2 inches.) v 1, f ast = v 1,slow = v 2, f ast = v 2,slow = (c) Use those velocity vectors to calculate the average acceleration for each circle. How big is each acceleration (in units of your sketch, ie., inches or centimeters)? a f ast = a slow = (d) State in words why the speed must affect the magnitude of the acceleration. 68
7 2 KINEMATICS 2.9 Motion in Two Dimensions 69
8 2.9 Motion in Two Dimensions 2 KINEMATICS 15. In a textbook or online, you ll find a derivation that shows that the acceleration of an object moving in a circle with radius r at constant speed v is equal to v 2 /r. (a) Are the units of this correct? (b) Is the formula qualitatively consistent with your results? That is, what does the formula predict when the radius doubles, and what was your result? How about when the speed doubles? 70
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