# Two balls in a bus

1. Aug 4, 2014

Hi
I had a question based on inertia
if two balls one heavy and one light are on a bus which is moving with velocity v and the bus stops, which ball will move faster and why?

2. Aug 4, 2014

### A.T.

Move faster relative to the bus?
Are the balls rolling or sliding relative to the bus?
If rolling, how do their moments of inertia compare?

3. Aug 4, 2014

See the bus has come to rest. So the balls will move forward due to inertia of motion. I am asking which will move faster

4. Aug 4, 2014

It depends.

5. Aug 4, 2014

### Staff: Mentor

If the balls are sliding not rolling, if friction is small enough that we don't have to worry about it, and everything is moor or less the same... Then both balls will move forward at the same speed.

This problem is easiest to understand if you think about from the point of view of someone sitting by the road watching. He sees a bus and two balls moving by at speed $v$. The bus stops but inertia keeps the two balls moving forward at the same speed.

If the two balls are rolling instead of sliding, or if we cannot ignore the effects of friction, the problem becomes much more complicated. Now the bus is exerting forces on the balls, and we have to understand these forces before we can say how they'll affect the two balls. That's why A.T. asked for more details.

6. Aug 5, 2014

Oh i see. Thanks for explaining it to me. We'll the balls are rolling not sliding now can u tell me the answer.

7. Aug 5, 2014

We'll they re rolling. What do u mean by moments of inertia?

8. Aug 5, 2014

### Staff: Mentor

If they start rolling across the floor of the bus, it's because friction between the floor of the slowing bus and the balls that are still moving forward at the same speed is making the balls rotate as they move forward. That force acts to slow the ball down even as it starts to rotate.

The "moment of inertia" is a quantity that tells us how much force and energy is needed to rotate an object, and it is [strike]greater for a heaver object[/strike] in general going to be different for the two objects. You need to know this to calculate exactly how the rolling objects will behave - which is why you've been getting so many "it depends" sorts of answers for the more complicated cases.

[edit - fixed the problem A.T. points out below]

Last edited: Aug 5, 2014
9. Aug 5, 2014

### A.T.

10. Aug 5, 2014

### A.T.

No, it's not necessarily greater for the heavier ball. That's why you need to know both, the masses and the moments of inertia to tell which ball will roll faster.

11. Aug 5, 2014

### Staff: Mentor

Thank you, you're absolutely right (and I think the rest what I wrote is consistent with the correct interpretation). I started it writing the answer one way assuming specific mass distributions and shapes, realized that the answer was getting too complicated, failed to completely clean it up before clicking submit.

Last edited: Aug 5, 2014
12. Aug 6, 2014

### sophiecentaur

Did anyone make the point that they are both travelling at the speed of the bus, just before it stops? Thereafter, the rate that they will be slowed down will depend upon the mass, MI and the friction situation (sliding / rolling).

13. Aug 6, 2014

### Buckleymanor

It's not as straight forewards as it looks.The balls are not in a gravitational freefall situation so depending on there mass they won't accelerate or deccelarate at the same speed.
Imagine a bowling ball and a ping pong ball then apply the same amount of force to each to accelerate them and bring them to a stop horizontaly.Taking into account friction and mass the bowling ball will move slower as the bus comes to a stop.

14. Aug 6, 2014

### jbriggs444

But one would not be applying the same force to each one in this scenario. As has already been pointed out, the situation at hand is equivalent to rolling the balls down an inclined plane.

Assuming that the coefficients of friction high enough so that the balls roll without slipping, neglecting rolling resistance and air resistance, total mass is obviously irrelevant. Scale is also irrelevant. [Double the radius and you've quadrupled the moment of inertia. But you've also doubled the moment arm and halved the rotation rate. The result is an unchanged resistance to rolling linearly down a ramp]

The only thing left that matters is mass distribution. A ping pong ball is hollow. That makes it resist rolling more strongly than a solid sphere like a bowling ball. Accordingly, the ping pong will tend to be decelerated more strongly as the bus brakes. Equivalently, it will roll down the ramp more slowly.

15. Aug 7, 2014

### sophiecentaur

Not exactly , because there is no gravitational force - but the idea of ignoring the bus bit is good!! (Simplify where possible). It is really an impulse problem and the simplest model would have no slippage between ball and floor. It strikes me that, as the bus has infinite mass and if you let it stop instantly, the initial peripheral speed of both balls must both be the same -i.e. the speed of the bus. I can't think of a mechanism that would make them different. Is this very simple explanation flawed?

16. Aug 7, 2014

### A.T.

The inertial force in the frame of the bus is of exactly the same form as the local gravitational force, or its component along an incline. One can view the inertial force in the frame of the bus as artificial gravity. So jbriggs444 is right that the faster ball in the bus will also be the faster one on an incline.

17. Aug 7, 2014

### sophiecentaur

I was assuming the bus stopped instantly. The OP does not say 'Slows down' and I took 'stopping', literally. But you are right, of course, if the bus has uniform acceleration.