Volleyball on the train

Classical Mechanics Level 2

Two trains are moving on parallel rails in opposite directions: the first has speed $5~\mbox{m/s}$ and the second has speed $7~\mbox{m/s}$ . A boy in the second train tosses a ball towards a boy in the first train at the moment they are facing each other. How fast should the first boy run in m/s with respect to the first train so that he can catch the ball?

Details and assumptions

Neglect the effects of gravity.

The answer is 12.

3 solutions

Michael Tang
Sep 19, 2013

We will consider only the horizontal component of the ball's path (the component parallel to both trains). Each second, the ball moves $7$ meters (due to its inertia from being in the second train), and the first boy moves $5$ meters in the opposite direction as the ball (because he is in the first train). Therefore, to be able to catch the ball, the first boy needs to move $7 + 5 = \boxed{12}$ meters per second horizontally, with respect to the first train.

Nicely Done !

Gabriel Merces - 7 years, 8 months ago

Kauan Santos
Sep 15, 2013

If the trains are in opposite directions and in one time the two boys get face to face and the trains have these speed, it's obviously that the ball should be with 12 m/s.

"Obvious" is an example of some dangerous words in solution-writing. I wish I had made my solution public. The reasoning here is good -- the intention is great. But this does not explain anything.

Here's a mathematical reasoning of what you are saying:

Horizontally, if it takes a time $t$ for the ball to get passed, in seconds, and we work with speeds in $\text{m/s}$ , the horizontal distance traveled by the ball is $7t$ , from the scenario's description in relation to the second train.

Now, we note that it is also $vt - 5t$ . Why? Because $v$ is the speed of the boy with respect to the ground, but we also have to subtract $5t$ , since the train contributes the negative of $5t$ .

Thus, $7t = vt - 5t \implies v = 12$ .

The same thing can be said conceptually.

Ahaan Rungta - 7 years, 9 months ago

I can't explain this way, it's awesome !

Kauan Santos - 7 years, 8 months ago

Thanks for the compliment! I'm glad to be able to help.

Ahaan Rungta - 7 years, 8 months ago

Shivam Gulati
Sep 20, 2013

We will consider only the horizontal component of the ball's path (the component parallel to both trains). Each second, the ball moves 7 meters (due to its inertia from being in the second train), and the first boy moves 5 meters in the opposite direction as the ball (because he is in the first train). Therefore, to be able to catch the ball, the first boy needs to move 7+5=12 meters per second horizontally, with respect to the first

Hmm... it seems a little similar to my solution. >:(

Michael Tang - 7 years, 8 months ago

Interestingly, his profile's first activity was in the same hour of copying your solution. I'm guessing he saw your solution, was enchanted by its beauty and decided to repost it, not copying the last word on purpose to prevent people from noticing.

Tim Vermeulen - 7 years, 8 months ago

How would he see the solution? Hmm.

Michael Tang - 7 years, 8 months ago

Volleyball on the train

The answer is 12.

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