Suppose a 10-kilogram ball and a 100-kilogram ball fall down in a straight line, which ball hits the ground first and why? We had a riot about this at room!

Answer

Hi Thibaut,

You have to start from Newton’s law: F=ma with a the acceleration and F the sum of all forces. All forces here are gravity F_z and the frictional force exerted by the air F_w .This frictional force is dependent on a number of factors: speed, shape, size,.. but is not dependent on the mass of the object.

So, if both balls are the same size and shape, then the heavy ball (100kg) will undergo a total force of 100×9.81 N+F_wso that the sphere has an acceleration of 9.81m/s²-F_w/100kg. The minus sign is there because the frictional force is of course opposite to the direction of movement.

The light ball (10kg) has the same acceleration of 9.81m/s²-F_w/10kg whereby we initially assume that both F_ware equal. It follows from this that the friction plays a smaller role with the heavier ball so that it has the greatest acceleration.

A complication here is that the frictional force is proportional to the speed squared so that the frictional force increases as the speed increases until at a certain moment the speed has become so high that the accompanying friction exactly counteracts gravity, so there is no more acceleration. and the ball continues to fall at a constant speed.

You can calculate this constant speed by F_w=av²=mg (with a constant depending on shape, size,…). This shows that the final velocity is greater when the sphere weighs heavier.

Conclusion: The heaviest ball will be down first as its acceleration is greater from the start and its limit speed also turns out to be greater than that of the lighter ball.

I hope you’re never arguing at room again! 🙂

Regards,

Lieven