• # Question: If you had two spheres of the same diameter and total mass, but one was hollow and the other was not, which would travel further along level ground if you gave them the same initial force?

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Asked by strangeness to Arttu, Ceri, James_M, Monica, Philip on 20 Jun 2011.
• Arttu Rajantie answered on 20 Jun 2011:

The filled one would travel faster and therefore further, because it has a smaller moment of inertia. In simpler terms, you can see that the atoms that are in the centre of the ball are moving along a straight line. The ones on the surface have to circle around the centre of the ball while the ball rolls, which means that they have to move a longer distance. Therefore they have to move faster in order for the whole ball to move at the same speed. The hollow ball only has these particles on the surface, and therefore with the same kinetic energy it will roll move slowly than the filled ball.

• James M Monk answered on 20 Jun 2011:

I think the answer is not as clear cut as that! It depends on the frictional forces the ball experiences.

If there is zero friction between the ball and the surface it is on and you apply the initial force (in fact we want to talk about impulse = force * time) directly to the centre line of the balls then they will both travel the same distance because there will be no rotation of the balls – they will just slide across the surface like a stone on a curling rink.

On the other hand, if there is friction with the surface and the force is applied slightly off-centre then the balls will rotate, so the hollow ball will have a higher angular momentum and less forward momentum. This means the air-speed over the hollow ball may be slower than the filled ball. Friction in air is proportional to the speed squared, so the hollow ball may lose less energy to air resistance because it is moving through the air more slowly – the hollow ball will travel further. On the other hand, this relies on the friction with the surface allowing the hollow ball to convert its rotational momentum back into forward momentum – if there is slippage on the surface (like tyre-skidding) then it will again lose energy.

So my answer is “it depends.” If both the friction with the air and the friction with the surface are high then the hollow ball travels further. If you make either one (or both) of the frictions low enough then the filled ball may travel further. It’s an interesting question.