# Velocity at impact

I am a high school student and have been wondering about this for a while: When a ball is either smashed or thrown down onto the ground, does the momentary velocity become zero at the moment of contact with the ground, similar to when a ball reaches its highest point when thrown vertically into the air? While we know that the velocity changes sign from negative to positive when the ball changes direction, I would like to understand whether the velocity graph crosses the $$x$$-axis and reaches zero at any point during this transition.

• Are you asking about the intermediate value theorem and the Bolzano theorem: en.wikipedia.org/wiki/Intermediate_value_theorem? That they apply follows from the assumption in the theory that the velocity is a continuous function of time (i.e. there are no infinite accelerations). Jun 3 at 16:28

Yes. You can either argue from the need for the velocity or momentum to be a continuous function of time, or the fact that the ground will, at some time, have imparted enough momentum upwards on the ball to just enough cancel its incoming momentum, before the next half of the momentum, needed to raise the ball back up to a new maximum height, is imparted.

Note that in this case, we do have to be careful, because the ball itself is undergoing internal motion when this is happening, and so we only mean that the centre of mass of the ball is momentarily stationary. The rest of the ball is in complicated motion at that time.

Typically a ball is idealized as a point particle or rigid object. A collision is idealized as instantaneous. The idealizations work for problems where you ask about velocity before and after. But they fail when you ask about what goes on during a collision.

Forces are reasonably small before a collision, typically $$1$$ g. A ball can stay reasonably rigid under forces like this.

A collision changes the velocity to $$0$$ in a very short time (usually). Accelerations and forces are very large. It isn't simple to analyze. Different parts of the ball can move at different speeds. If you do analyze it, every part decelerates to $$0$$ in a continuous way. If the collision is elastic, every part accelerates back up to speed.

Here is an example of what goes on during a collision - Titleist Golf Balls at The Moment of Impact. This one is more fun - How Hard Can You Hit a Golf Ball? (at 100,000 FPS)

Does the momentary velocity become zero ?

Yes, as long as you assume a physically realistic scenario in which the ball and/or the ground deforms (even if only slightly). In this case the velocity of the ball (more precisely, the velocity of its centre of mass) will be a continuous function of time. Since this velocity changes sign during the collision with the ground, there must be some point in time while the ball is in contact with the ground when its velocity is instantaneously zero.

However, this conclusion does not hold if you make the (unrealistic) assumption that both the ball and the ground are perfectly rigid. In this scenario, the velocity of the ball is not a continuous function of time, and it can change sign without ever having a value of zero.

In real life, yes it does! If this wouldn't have been the case, then the ground is supposed to provide an infinite amount of force to the ball which is not possible in real life scenarios. Moreover the ground also deforms a bit in real life which hints towards a continuous change of velocity.

The graphs ( you are talking about) assume a perfect ideal system in which the mass of the ground (earth) is infinite and collisions are perfectly elastic.