Chapter 4, Problem 24A

A ball is released from rest at the left end of three different tracks. The tracks are bent from pieces of metal of the same length.

a. From fastest to slowest, rank the tracks in terms of the speed of the ball at the end. Or, do all balls have the same speed there?

b. From longest to shortest, rank the tracks in terms of the time for the ball to reach the end. Or do all balls reach the end in the same time?

c. From greatest to least, rank the tracks in terms of the average speed of the ball. Or do the balls all have the same average-speed on all three tracks?

To determine

To rank: The tracks in terms of speed of ball at the end from fastest to slowest.

Answer to Problem 24A

B > C = A

Explanation of Solution

Given:

The three different tracks are shown below.

Also, ball is released from rest position.

Formula used:

Assuming there is no friction on the tracks. When the ball is moving on the downfall curve then speed increases due to gravitational force. When it moves its upward curve then speed decreases as the ball has to do work against gravitational force.

Calculation:

On track A, only 1 downward curve is present. So, the ball will gain speed and then moves in constant speed on the straight line.

On track B, there are 2 downward curves present. So, the ball will gain speed with the first one and then after moving for a while in a straight line will again gain speed on the second downfall. Thus, end up with the maximum speed.

On track C, there are 2 downfalls and one upward curve. The ball will first behave as it was on track B and then due to the upward curve, it will work against gravity. Thus, its speed will decrease and ends up with similar speed as the ball has at the end of track A.

Conclusion:

Thus, balls have equal speed on tracks C and A. But, on track B, ball has maximum speed. Hence, ranking will be such as B> C= A.

To determine

To rank: The tracks from longest to shortest in terms of time taken by the ball to reach the end.

Answer to Problem 24A

A> B> C.

Explanation of Solution

Given:

The three different tracks are shown below.

Also, ball is released from rest position.

Formula used:

Time is inversely proportional to speed for a particle for a distance.

Calculation:

Consider track A. The ball in this case will accelerates when it is on the downward curve. After that it travels with a constant speed. Thus, travelling with the constant speed will have the longest time taken of all three.

Consider track B. During each downward curve, the ball will accelerate.This means that it will accelerate twice. So the ball will end up with the higher speed. Falling twice on the downward tracks will have the maximum speed. So, it quickly speeds up and thus lesser time will be taken to reach to the end.

Consider track C. The ball will accelerate twice, but during the upswing segment it will decelerates once. So its velocity on the final segment is same after the dip, as it was before.

C track can be illustrated as shown in the figure below.

P: slower, but accelerating

Q: faster and constant speed

R:firstly, accelerates and moves with constant speed then decelerates

S: fastest speed equivalent to segment Q

The average speed of the ball will have faster speed as on segment Q. Thus, will take the least amount of time to get to the end.

Conclusion:

Therefore, the rank is A>B> C

To determine

To rank: The tracks in terms of average speed of the ball from greatest to least.

Answer to Problem 24A

Ranks from largest to the least average speed of the ball is as follows C> B> A.

Explanation of Solution

Given:

The three different tracks are shown below.

Also, ball is released from rest position.

Formula used:

Average speed is calculated as the total distance travelled on the tracks divided by the time taken.

Calculation:

Time and speed are inversely proportional to each other. So, for longest time average speed will be less in comparison to the shortest time for the same distance.

From part (b), time taken by ball on track C is lowest so the average speed will be highest. While the time taken by ball on track A is highest so the average speed will be lowest

Conclusion:

Therefore, rank of tracks on the basis of average speed is C> B> A.