# Homework Help: Kinematics Rocket Question

1. Jun 18, 2014

### sankalpmittal

1. The problem statement, all variables and given/known data

A rocket is moving in a gravity free space with a constant acceleration of 2 m/s2 along + x direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in + x direction with a speed of 0.3 m/s relative to the rocket. At the same time, another ball is thrown in -x direction with a speed of 0.2 m/s from its right end relative to the rocket. The time in seconds when the two balls hit each other is ?

Image: http://postimg.org/image/fh9trs9r5/

The answer ranges from 0 to 9.

2. Relevant equations

http://en.wikipedia.org/wiki/Relative_velocity

3. The attempt at a solution

Here is what I did.

If rocket accelerates by 2 m/s2 in +x direction, then balls inside it also accelerates with it with same value and in same direction in ground frame. Thus when we switch to frame of the rocket there is no acceleration on balls. Inside that frame I impose the concept of relative velocity and convert this two body problem to one body.

Velocity of left ball relative to right ball = 0.5 m/s in +x direction.

Thus time taken to collide = distance/Velocity of left ball relative to right ball = 4/0.5=8 seconds.

However exam in which this question was asked says that both 8 seconds and 2 seconds answers are acceptable.

One book offers solution as :

Maximum displacement of the left ball from the left wall of the chamber is 2.25 cm, so the right ball has to travel almost the whole length of the chamber (4m) to hit the left ball. So the time taken by the right ball is 1.9 sec (approximately 2 sec).

Now my doubt is as follows :

How is 2 seconds a correct answer ? Also how can there be "two" values of time of collision ?

Last edited: Jun 18, 2014
2. Jun 18, 2014

How do you know that? If the ball is thrown from one end, then why should it accelerate along with the rocket?

3. Jun 18, 2014

### sankalpmittal

Observe from ground frame yourself.

See, if you are sitting in an accelerated train suppose, and you throw a ball vertically, it comes back to your hand right ? This is because it is accelerated with the train.

Waiting for someone else to reply though.

4. Jun 18, 2014

I know that. I actually forgot about the "relative" word in your question

5. Jun 18, 2014

### sankalpmittal

No problem.

Just waiting for someone to reply and help me. xD

It's confusing question..

6. Jun 18, 2014

### matineesuxxx

I think Im going to have to refute that. As long as the ball is in your hand while you are on the accelerated train, then yes it is accelerating, but as soon as you throw it what forces are acting on it? None. The train keeps accelerating due to some force, but the ball being in air is subject only to its initial velocity and g with respect to the ground, no? If the train is going at a constant velocity, then the relative velocity is zero, yet if the trian is accelerating with acceleration $\hat{a}$, then the balls acceleration relative to the train is $-\hat{a}$.

EDIT: Remember: $\hat{a}_{\text{1,2}} = \hat{a}_{\text{1,g}} - \hat{a}_{\text{2,g}}$

Last edited: Jun 18, 2014
7. Jun 18, 2014

### Staff: Mentor

This problem can be solved for either the rocket frame of reference or for the fixed frame of reference. Both approaches will give the same answer. Which frame do you prefer?

Chet

8. Jun 18, 2014

### sankalpmittal

Hello Chet ! :)

I would like to solve the problem with both approaches. First I will do rocket frame approach. But do you not think I already solved it in my attempt at solution using rocket frame ?

Last edited: Jun 18, 2014
9. Jun 18, 2014

### Staff: Mentor

In the rocket frame of reference, the system behaves as if there is a gravitational acceleration of 2 m/s2 in the negative x-direction. So, for the ball thrown from the left end at x = 0, the ball's location at time t is xl = 0.3t - t2. For the ball thrown from the right end at x = 4, its location at time t is xr=4 -0.2t - t^2. The balls meet when xl=xr.

time left ball right ball

0 0 4
1 -0.7 2.8
2 -3.4 -0.4
3 -8.1 -5.6
4 -14.8 -12.8
etc.

These results make no sense. Are you sure about that acceleration? Could it be 0.2 m/s2? With this acceleration, the left ball falls back to the left wall in a very short time.

Chet

10. Jun 18, 2014

### Staff: Mentor

The left ball falls back to the left wall every 0.3 seconds, and it never gets further from the wall than 0.0225 m, even if it bounces elastically. So the question really boils down to how long it takes for the right ball to fall to ~ 0.

Chet

11. Jun 18, 2014

### Tanya Sharma

Hi Sankalp

Nice question :)

This approach is not correct because as a result of collision of the left ball with the left wall of the rocket ,the velocity of the left ball changes after every 0.3 seconds (assuming elastic collision) .

I think 8 seconds cannot be the correct answer .OTOH approx. 2 seconds looks alright.

12. Jun 19, 2014

### ehild

NO.

It is true if the train travels with constant velocity.

In an accelerating train, the released ball moves backward. Just like you if you stand and do not grab something fixed.

ehild

13. Jun 19, 2014

### ehild

Very good!

ehild

14. Jun 19, 2014

### sankalpmittal

Thanks to all. :)

I think I figured it out.

From the rocket frame we have acceleration of 2 m/s2 in -x direction on each ball.

So maximum displacement traveled by left ball is u2/2a = 0.025 m.

Note that acceleration is still acting on left ball in -x direction.

Time taken :

0=0.3-2t
t=0.15 s.

In this time:

Distance by right ball :

-s=-0.2t-t2

s= 0.05 m.

Now 0.08 + 0.05 <<4

The right ball has to traverse a distance of 4-x, where x<0.02..

Hence approximating 4-x≈4 we have

-4=-0.2t-t2

This gives t=1.9 s ≈ 2 s...

Thanks to Chet.. His third post explains the reason for this theoretically.

Now why exam accepted 8 s ?

That's my reasoning :

If the balls are "kept on the floor of rocket cabin", then they have the same acceleration as rocket. Hence from rocket frame there is no acceleration on balls. Inside that frame I impose the concept of relative velocity and convert this two body problem to one body.

Velocity of left ball relative to right ball = 0.5 m/s in +x direction.

Thus time taken to collide = distance/Velocity of left ball relative to right ball = 4/0.5=8 seconds.

Correct ?

Last edited: Jun 19, 2014
15. Jun 19, 2014

### Tanya Sharma

Thanks

16. Jun 19, 2014

### Tanya Sharma

That is still the same situation as in OP .Keeping the balls on the floor cabin do not give the balls same acceleration as that of rocket.

I think there is only one correct answer and that is 2 seconds.

What is 0.02 on the LHS ?

17. Jun 19, 2014

### sankalpmittal

Why ? When you are sitting in the train, you are accelerated with the train. When you throw the ball up, then that acceleration doesn't count with it.

I can't say that 8 s cannot be the answer.

0.02 is the distance traversed by the left ball when it velocity becomes zero due to deceleration.

18. Jun 19, 2014

### Tanya Sharma

OK.

Assuming frictionless surface ,balls motion would be independent of the motion of the rocket .

In the train example ,when the person throws the ball up ,the acceleration of the ball will still be 9.8ms-2 . But if you are working from the train reference then surely you need to take into account the pseudo force .

But accounting for pseudo force is not the same thing as if the ball is having horizontal acceleration of the train in addition to its vertical gravitational acceleration.

If 2 sec is the correct answer ,what does 8 seconds signify ?

19. Jun 19, 2014

### sankalpmittal

More to the point.

8 s should signify that the right ball is taking approx. 6 seconds to move from left to right and then again left.

It has already taken 2 s to move from right to left initially.

Assume that left ball is fixed at x=0 for the sake of approximation.

Correct ?

Last edited: Jun 19, 2014
20. Jun 19, 2014

### Tanya Sharma

Have you calculated the time it takes for the right ball to move from left to right end ,and left again ?

Does it come out to be 6 sec ?

I am curious to know what makes you so sure that 8 seconds is the correct answer .The answer key may be wrong .Which book are you referring to ?

21. Jun 19, 2014

### sankalpmittal

Nope :(

Actually it comes out to be 3 seconds and the final answer as 3+2=5 seconds :(

Now why is 5 s not the answer ?

22. Jun 19, 2014

### consciousness

See there are two possibilities-

i)Collisions between the ball and the rocket don't occur (the rocket is assumed to be very long or whatever)

In this case the answer is found very easily to be 8 sec. (Just see the relative speeds of the balls and divide them by the distance between them)

ii)Collisions occur (seems more reasonable when we look at the figure)

This raises further questions. Specifically, what is the coefficient of restitution for the collision? Since this information isn't given, we cannot proceed further.

So the question is incorrect.

23. Jun 19, 2014

### Tanya Sharma

2 seconds is the correct answer . Wherever 8 seconds is published ,it is surely done hastily assuming that the left ball doesn't collide with the left wall ,which is an incorrect assumption.

It is implicit that the question asks for the time when the two balls collide for the first time and you agree that it is 2 seconds .Why should you be concerned about the time when they collide for the second or third or nth time ?

24. Jun 19, 2014

### sankalpmittal

If they would have said minimum time taken for collision I would have agreed without questioning that it is 2 seconds and nothing more than that.

Nope. It is a 3 body problem. If you see from rocket frame you are bringing rocket "virtually" to rest and this imposes acceleration of 2 m/s2 on each of the ball in -x direction.
You can do nothing more than that. You seem to be thinking that we find relative velocity of one ball w.r.t another. Albeit, that doesn't mean that you ignore acceleration of rocket.
You can't convert this 3 body problem to one body.

Well you are right here but everyone will assume perfect elastic collision. I hope you have given this paper.

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Last edited: Jun 19, 2014
25. Jun 19, 2014

### Staff: Mentor

Probably the reason they accepted 8 seconds on the exam was that they felt that the students did not have time to reason it out in detail correctly. So they gave them a break. But, as Tanya pointed out, the only real correct answer is ≈ 2 sec.

Chet