Assignment 4 is due Thursday, January 29 Name: Key Average = 7.9

Train A, full of eager and enthusiastic observers, is moving east with a velocity of 50 ft/s with respect to the station. Train B, also full of alert passengers, is moving west with a velocity of - 50 ft/s with respect to the station. Note that we are using the same sign convention as the text, speeds towards the east correspond to positive velocities while speeds towards the west correspond to negative velocities.

All the action in this assignment takes place as the two trains race through the station. And yes, you guessed it, the station and trains are populated with skillful and astute observers!

There are three basic collisions that are going to be analyzed by the different sets of observers.

Case 1: Two perfectly elastic balls, one blue and other red, with identical masses head towards each other with equal speeds, for example 25 ft/s. For concreteness imagine that the blue ball is heading east, Vblue = + 25 ft/s, while the red ball is heading west,

Vred = - 25 ft/s. After the head-on collision, the blue ball moves to the west at 25 ft/s while the red ball heads east also at 25 ft/s. In this reference frame where the balls have equal and opposite speeds the results of the collision are particularly easy to understand.

Case 2: Imagine the same situation as in Case 1 except that the balls are made of clay and stick to one another after the collision. For two clay balls of equal mass going in opposite directions with identical speeds, the collision produces a red and blue blob at rest, velocity of zero.

Case 3: This is the most interesting of the three cases. Here we have a Aheavy@ perfectly elastic blue ball colliding head on with a Alight@ red ball. When observed from a reference frame in which the Aheavy@ ball is stationary, the Alight@ ball bounces back with the same speed it had before the collision. For example, if the Alight@ ball were moving with a velocity of + 40 ft/s before the collision, it moves with velocity - 40 ft/s after the collision. The Aheavy@ ball remains stationary during the whole collision process. This collision is between a Alight@ and Aheavy@ ball only if the Aheavy@ ball is stationary before, during, and after the collision!

Chapter 1 has examples where the above basic cases are used to understand collisions in which the initial velocities in Cases 1 and 2 are not equal and opposite and in situations with the heavy ball in Case 3 is not stationary. Viewing a variety of physical situations from different reference frames is an essential part of coming to grips with special relativity.

In all the following scenarios, the station observers measure the initial velocity of the blue ball to be + 40 ft/s while the initial velocity of the red ball is - 20 ft/s. The blue ball is moving toward the east and the red ball toward the west in the reference frame of the station. Also, please put all your answers into the table on the last page.

The first 3 exercises refer to the initial velocity of the red and blue balls.

Exercise 1.

The answers to these two questions are as obvious as they appear, no tricks!

From the perspective of observers on the station, the velocity of the blue ball = +40 ft/s .

From the perspective of observers on the station, the velocity of the red ball = -20 ft/s .

Exercise 2.

From the perspective of the observers on train A moving to the east at 50 ft/s, what are the velocities of the blue and red balls? Think carefully about how the balls appear to move when viewed through the windows of the train racing toward the east at 50 ft/s.

From the perspective of observers on train A, the velocity of the blue ball = -10 ft/s .

From the perspective of observers on train A, the velocity of the red ball = -70 ft/s .

Exercise 3.

From the perspective of the observers on train B moving to the west at 50 ft/s, what are the velocities of the blue and red balls? Think carefully about how the balls appear to move when viewed through the windows of the train racing toward the west at 50 ft/s.

From the perspective of observers on train B, the velocity of the blue ball = + 90 ft/s .

From the perspective of observers on train B, the velocity of the red ball = + 30 ft/s .

Collision 1.

The blue and red balls have the same mass and are perfectly elastic. As observed in the station, the blue ball has a velocity of + 40 ft/s and the red ball has a velocity of -20 ft/s.

Exercises 4 through 6 refer to the velocities of the balls after collision 1.

Exercise 4.

What are the velocities of the two balls in the reference frame of the station after the collision? (Hint: Imagine the collision takes place in a frame where the two balls have equal and opposite speeds since we know the results in that special case. Then switch your point of view back to the station. Easy to say, not so easy to do!)

From the perspective of observers on the station, the velocity of the blue ball after the collision = -20 ft/s .

From the perspective of observers on the station, the velocity of the red ball after the collision = + 40 ft/s .

Exercise 5.

What are the velocities of the two balls in collision 1 in the reference frame of train A after the collision? You ought to be able to use your answers to Exercise 4 to get these answers.

From the perspective of observers on train A, the velocity of the blue ball after the collision = - 70 ft/s .

From the perspective of observers on train A, the velocity of the red ball after the collision = -10 ft/s .

Exercise 6.

What are the velocities of the two balls in collision 1 in the reference frame of train B after the collision? You ought to be able to use your answers to Exercise 4 to get these answers.

From the perspective of observers on train B, the velocity of the blue ball after the collision = + 30 ft/s .

From the perspective of observers on train B, the velocity of the red ball after the collision = + 90 ft/s .

Collision 2.

The blue and red balls have the same mass and are sticky clay blobs. As observed in the station, the blue ball has a velocity of + 40 ft/s and the red ball has a velocity of -20 ft/s.

Exercises 7 through 9 refer to the velocity of the blob after collision 2.

Exercise 7.

What is the velocity of the blob in the reference frame of the station after the collision? (Hint: Imagine the collision takes place in a frame where the two balls have equal and opposite speeds since we know the results of that special case. Then switch your point of view to the station.)

From the perspective of observers on the station, the velocity of the blob after the collision = + 10 ft/s .

Exercise 8.

What is the velocity of the blob in the reference frame of train A after the collision? You ought to be able to use your answer to Exercise 7 to get the answer.

From the perspective of observers on train A, the velocity of the blob after the collision = -40 ft/s .

Exercise 9.

What is the velocity of the blob in the reference frame of train B after the collision? You ought to be able to use your answer to Exercise 7 to get the answer.

From the perspective of observers on train B, the velocity of the blob after the collision =

+ 60 ft/s .

Collision 3.

The Aheavy@ blue ball and the Alight@ red ball have a perfectly elastic collision. As observed in the station before the collision, the blue ball has a velocity of + 40 ft/s and the red ball has a velocity of -20 ft/s.

Exercises 10 through 12 refer to the velocity of the balls after collision 3.

Exercise 10.

What are the velocities of the two balls in the reference frame of the station after the collision? (Hint: Imagine the collision takes place in a frame where the Aheavy@ blue ball is stationary since we know the results of that special case. Then switch your point of view to the station.)

From the perspective of observers on the station, the velocity of the blue ball after the collision = +40 ft/s .

From the perspective of observers on the station, the velocity of the red ball after the collision = + 100 ft/s .

Exercise 11.

What are the velocities of the two balls in collision 1 in the reference frame of train A after the collision? Remember, the collision is described in terms of velocities in the station frame. You ought to be able to use your answers to Exercise 10 to get these answers.

From the perspective of observers on train A, the velocity of the blue ball after the collision = -10 ft/s .

From the perspective of observers on train A, the velocity of the red ball after the collision = + 50 ft/s .

Exercise 12.

What are the velocities of the two balls in collision 1 in the reference frame of train B after the collision? Remember, the collision is described in terms of velocities in the station frame. You ought to be able to use your answers to Exercise 10 to get these answers.

From the perspective of observers on train B, the velocity of the blue ball after the collision = + 90 ft/s .

From the perspective of observers on train B, the velocity of the red ball after the collision = + 150 ft/s .

Below are the tabulated answers for homework assignment 4.

Before the collision, the velocity of the balls in ft/s in the different frames are:
	Train A	Station	Train B
Blue Ball	-10	+40	+90
Red Ball	-70	-20	+30
After collision 1, the velocity of the balls in ft/s in the three frames are:
Blue Ball	-70	-20	+30
Red Ball	-10	+40	+90
After collision 2, the velocity of the combined ball in ft/s in the three frames is:
Red/Blue Blob	-40	+10	+60
After collision 3, the velocity of the balls in ft/s in the three frames are:
Blue Ball	-10	+40	+90
Red Ball	+50	+100	+150