## Question:

Newton’s cradle is a common desk toy consisting of a number of identical pendulums with steel balls as bobs. These pendulums are arranged in a row in such a way that, when at rest, each ball is tangent to the next and the cords are all vertical. Assume that the COR for the impact of a ball with the next is e = 1. Explain why if you release the two balls to the far left from a certain angle, the balls in question come to a stop right after impact while all the other balls do not seem to move except for the two balls to the far right, which swing upward and achieve a maximum swing angle equal to the initial release angle of the two balls to the far left.

## Step-by-step

Treat each impact as only involving two balls. The first impact will occur between balls 2 and 3. Because the COR e = 1 and the masses are identical we see from the solution to Problem 5.47 that ball 2 will come to a complete stop after impacting with ball 3. We also see that ball 3 will have a post impact velocity identical to the pre impact velocity of ball 2. Call this velocity ${ \upsilon }_{ 0 }$. At the same instant ball 2 impacts ball 3 ball 1 impacts ball 2. Ball 1 will stop and ball 2 will have a post impact velocity ${ \upsilon }_{ 0 }$. Now balls 2 and 3 have velocity ${ \upsilon }_{ 0 }$. When ball 3 impacts ball 4 ball 3 stops and ball 4 has a post impact velocity ${ \upsilon }_{ 0 }$, ball 3 is impacted by ball 2, ball 2 stops and ball 3 has a post impact velocity ${ \upsilon }_{ 0 }$. Now balls 3 and 4 have velocity ${ \upsilon }_{ 0 }$. When ball 4 impacts ball 5 ball 4 stops and ball 5 has a post impact velocity ${ \upsilon }_{ 0 }$, ball 4 is impacted by ball 3, ball 3 stops and ball 4 has a post impact velocity ${ \upsilon }_{ 0 }$. The work-energy principle tells us that balls 4 and 5 will stop moving when they have reached the initial height balls 1 and 2 were released from. Finally, since the lengths of the pendulums are identical the maximum swing angle of balls 4 and 5 are equal to the initial release angle of balls 1 and 2.