F FA ACULTY C U L T Y OF O F EDUCATION E D U C A T I O N Departent of Curriculu and Pedagogy Physics Circular Motion: Energy and Moentu Conservation Science and Matheatics Education Research Group Supported by UBC Teaching and Learning Enhanceent Fund 0-03
Question Seicircles Title IV
Question Elastic Collisions Title This is a challenging set for the students who are interested in physics and like challenges. However, it is also a very beautiful set which will help students build their intuition. In order to be able to solve questions related to circular otion, the students have to know how to do collision probles. Therefore, we review elastic collisions in the first part of the set and then ove to discuss circular otion v v x
Question Perfectly Elastic Title Collisions I Two balls with asses of and are oving towards each other with speeds of v and v respectively. If the collision between the balls is a perfectly elastic collision, what will the speeds of the balls be after the collision? A u. C. u v v v v u v v v v u u B. D. v v v u v u v v v v u v v x
Coents Solution Answer: A Justification: We can find correct answer by applying the laws of energy and oentu conservation as shown below. However, since it is a ultiple-choice question, we can also attept to eliinate the wrong answers. Answers B-D all have incorrect units. You can see that one of the ters in the is squared thus aking the units in the nuerator or in the denoinator incorrect or not atching. v v u v v v u u u u v v v v
Question Perfectly Elastic Title Collisions II A ball with ass is oving towards another ball with ass. The second ball is initially at rest (v = 0). If the collision between the balls is a perfectly elastic collision, what will the speeds of the balls be after the collision? v u u v A. C. v u u v v v u u B. D. v v u u E. None of the above v v =0 x
Coents Solution Answer: B Justification: We can find correct answer by using the collision equation fro the previous question and assuing that v =0 (initial velocity of the second ball is zero): Since v 0 : u u v v v v v v Notice, the answers are not syetrical (which akes sense as we had asyetrical initial conditions. You can check that the correct answer also has correct units. In both cases, the denoinator represents the ass of the syste. The nuerator of the first equation represents the difference of asses, we will explore its eaning in the next question.
Question Perfectly Elastic Title Collisions III A ball with ass is oving towards another ball with ass. The second ball is initially at rest (v = 0). If the collision between the balls is a perfectly elastic collision and =, what will the speeds of the balls be after the collision? u 0 u v A. C. u v u v u v u v B. D. u v u 0 u 0 E. u v v v =0 x
This tells us that the ball that was oving initially will stop and the ball that was originally at rest will start oving with the initial speed of the first ball. This phenoenon is used in a faous deonstration called Newton s cradle: https://www.youtube.co/watch?v=0lnbyjoyeq8 You can see that none of the other answers ake sense! Coents Solution Answer: B Justification: We can find correct answer by using the collision equation fro the previous question and assuing that not only v =0 (initial velocity of the second ball is zero) but also that Assuing that initial velocity of the second ball is zero: v 0 and u u v v 0 v v v v
Question Perfectly Elastic Title Collisions IV A ball with ass is oving towards another ball with ass. The second ball is initially at rest (v = 0). If the collision between the balls is a perfectly elastic collision and >>, what will the speeds of the balls be after the collision? u v u v A. C. u v u v u v u v B. D. u v u 0 u v E. u v v v =0 x
Coents Solution Answer: D Justification: We can find correct answer by using the collision equation fro the previous question and assuing that not only v =0 (initial velocity of the second ball is zero) but also that or Since v 0 and : v v v u v v v v v u 0 This is a very interesting conclusion. While the heavy ball will practically stop, the light ball will bounce off it with the speed equal to its initial speed. This akes sense considering our earlier discussion. You can see that none of the other answers ake sense!
Question Perfectly Elastic Title Collisions V A ball with ass is oving towards another ball with ass. The second ball is initially at rest (v = 0). If the collision between the balls is a perfectly elastic collision and >>, what will the speeds of the balls be after the collision? u v u v A. C. u v u v u v u v B. D. u v u v E. u u v v v v =0 x
Coents Solution Answer: E Justification: We can find correct answer by using the collision equation fro the previous question and assuing that not only v =0 (initial velocity of the second ball is zero) but also that Since v 0 and : v v v u v v v v v u v or While the heavy ball will continue oving alost unaffected, the light ball that was initially at rest will bounce off with the speed equal to twice the speed of the heavy ball v. How coe? This is easier to understand in the frae of reference of ball. In that frae of reference, ball will be oving towards with the speed v and as we discussed earlier it will bounce off with the speed v relatively to ball. However, since ball is oving relatively to the ground with the velocity v, the velocity of ball relatively to the ground will be v.
Question Perfectly Elastic Title Collisions VI Balls and are oving towards each other with equal speeds v relatively to the ground. If the collision between the balls is a perfectly elastic collision and =, what will the speeds of the balls be after the collision? u v u v A. C. u v u v u v u v B. D. u v u v E. u u v v v = v v = -v v v x
Coents Solution Answer: A Justification: We can find correct answer by using the collision equation fro the previous questions and assuing that not the velocities of the balls are opposite and their asses are equal: v v v and u u v v v v v v v( ) v v v v v v It akes sense that the balls will bounce off each other and will ove in opposite directions with the sae speed as they had before the collision. You can see that none of the other answers ake sense!
Question Perfectly Elastic Title Collisions VII A ball with ass is oving towards a very big wall. If the collision between the ball and the wall is a perfectly elastic collision, what will be the result of the collision and what will happen to the wall? FIND THE WRONG STATEMENT: A. The ball will bounce back with the speed v B. The ball will bounce back with the speed v C. The wall will bounce back with the speed of v D. The ball and the wall is NOT a closed syste, so the oentu of the syste will not be conserved E. The wall will be copressed and it will exert a force on the ball that according to Newton s third law will be: F ball on wall F wall on ball v M x
Coents Solution Answer: B Justification: The only incorrect stateent is B. The wall is connected to the ground, so the ball and wall is NOT a closed syste. While the ball will not ove, it will exert a force on the ball that will ake the ball bounce back with the sae speed it cae with. Notice, the law of oentu conservation only works for two objects that interact with each other and are unaffected by other objects. In this case, the wall is affected by the ground. If the wall was on wheels, or was able to ove back, then the wall would have oved with the speed of v and the ball would have bounced back with the speed of v.
Question Part II: Circular Title Motion The questions in this sub-set cobine concepts of circular otion, energy and oentu conservation. v 0
Question Title Seicircles VIII A hollow and frictionless seicircle is placed vertically as shown below. A ball enters with an initial speed v 0 and exits out the other end. What is the final speed of the ball? A. v0 v0 B. r v C. 0 D. v0 3 E. No idea v 0
Coents Solution Answer: A Justification: Since the sei-circle is frictionless, the echanical energy of the ball has to be conserved. The entrance and exit points of the seicircle are at the sae height. This eans the gravitational potential energy of the ball is equal at these two points. Therefore, the kinetic energies ust also be the sae, and the speed of the ball is the sae upon entrance and exit of the sei circle.
Question Seicircles Title IX A hollow and frictionless seicircle is placed vertically as shown below. A ball enters with an initial speed and exits out the other end. If the seicircle has a radius r, what is the iniu value of v 0 that will allow the ball to coplete the path around the seicircle? A. gr B. gr C. 3gr D. gr E. g r v 0 r
Coents Solution Answer: C Justification: At the top of the seicircle, where the velocity is lowest, the centripetal force ust be greater than the force of gravity to prevent the ball fro falling. Applying Newton s second law and the law of energy conservation: Newton s second law v top gr (positive x is directed g N If N 0, vtop vtop gr r in upward) v v 0 gr top v gr 0 0 0 gr v gr gr v 3gr Energy conservation law for the point at the top of the trajectory. Potential energy at the botto of the trajectory is zero.
Question Seicircles Title X A contraption built out of two seicircular tubes is shown below. A ball with a ass of enters with an initial speed v 0 and elastically collides with a ball of ass. The ball with ass then exits out the other end. What is the final speed of the ball with ass if >>? A. v 0 / B. v 0 C. v 0 D. 3v 0 v 0 E. No idea
Coents Solution Answer: C Justification: We know fro question VIII that the speed of ball when it coes out of a sei-circle ust be v 0. So when collides with it will be travelling at a speed of v 0. Fro the reference frae of, travels towards it at a speed of v 0. Since is uch heavier than, will be alost unaffected by the collision (it will continue oving with the speed v 0 ), while will bounce off with a speed of v 0, since its velocity has to be v0 relatively to ball - (See part I of this set, questions IV and V). Because returns to the sae height after it has rounded the larger seicircle, it will travel at the sae speed it had before traversing the seicircle, which is v 0. Of course you can solve it algebraically, by using the equations for elastic collisions of two balls of different asses as we did earlier
Question Seicircles Title XI A. gr B. gr This is the sae situation as question X, where >>. What is the iniu value of v 0 required for to exit? C. 3gr D. gr E. 3gr 3r r v 0
Coents Solution Answer: C Justification: This question is soewhat trickier than the previous one. In the previous questions we found that the speed required for a ball to coplete a sei-circle is 3gR. Therefore, for the sall and big sei-circles, it is: v 3gr sall _ in v 3g3r 3v.7v v v l arg e_ in sall _ in sall _ in sall _ in 0 Since travels with v 0 and the iniu speed required to coplete a large sei-circle is less than twice the speed needed to coplete a sall sei-circle, the iniu speed required for to exit is not the speed required for to travel around the seicircle, but rather the speed required for to coplete the sei-circle and hit, as can already traverse the seicircle at that speed. Thus, the answer to this question has the sae answer as question X.
Question Seicircles Title XII A C. E. n. 3 n. 3 (5 ) n 3 g(5 r) n n. 3 (5 ) n 3 g(5 r) n gr B g r D g r n A contraption built out of seicircles is shown below. A ball with a ass of enters with an initial speed v 0. After passing the first seicircle, elastically collides with a ball with ass, which then rounds the second seicircle and elastically collides with a ball of ass 3 and so on in the fashion shown in question 4. If >> >> >> n and the n th seicircle has radius 5 n- r, what is the iniu value of v 0 for the n th ball to round its seicircle?
Coents Solution Answer: E Justification: We know fro questions III-V that travels at v 0 when hits it with speed v 0. As >> >> >> n, 3 travels at v if hits it with v. Therefore, n travels with a speed of n- v 0. At the n th seicircle, the radius is 5 n- r. Fro the identity we saw in question X, the iniu speed for the ball not to fall off a seicircle is v= (3gr). In our case, it translates into: v 3 g(5 r) v n n 0 3 g(5 r) n n n 0 5 3 n gr