Kinetics Question

[Badly Shaved Monkey]

While I'm in question asking mood, here's another one, just out of curiosity.

There was an advert on the radio talking about the current 6-Nations rugby and it mentioned the speed at which a rugby ball could be kicked. I've wondered about this before. When a ball is accelerated by impact (golf club swing, football kick, tennis racket impact) is the maximum velocity of the ball no greater than the velocity of the impactor or does the elasticity of the ball and impactor and general non-Newtonianness of the situation alter things so that the ball can spring away with a velocity higher than that of the incoming blow?

 

[Mojo]

When a ball is accelerated by impact (golf club swing, football kick, tennis racket impact) is the maximum velocity of the ball no greater than the velocity of the impactor or does the elasticity of the ball and impactor and general non-Newtonianness of the situation alter things so that the ball can spring away with a velocity higher than that of the incoming blow?

Would the mass of the ball relative to the object (club, bat or foot) hitting it be relevant as well?

 

[Crispy Duck]

Hmmm... I suspect the answer is yes. I've seen some ultra-slow-mo footage of a golf ball being struck, and the ball definitely sprang away from the club head, even as the club head was still following-through. As you say, this seemed to be due to the stored energy in the deformed golf ball - it deformed quite significantly as the club head first hit it.

At the point of maximum deformation, the situation could be analysed like this - the club head and the golf ball are both moving at the same speed, but the golf ball is significantly deformed, thus providing a significant force in *both* directions - the "elastic force" from the ball is pushing the ball forward and the club back.

This isn't at all 'non-Newtonian', of course, but I'm sure you knew that

 

[Crispy Duck]

See this rather wonderful site for some 5000fps videos:

http://www.engr.colostate.edu/~dga/high_speed_video/#Sports_Equipment

Re: other sports... in racket sports - tennis, squash etc - the racket itself also deforms significantly, which adds to the acceleration, I guess. But thinking of bowling, in cricket (or pitching in baseball, I guess), I would guess there's no significant deformation of either ball or hand, so to bowl at 100mph, a bowler's hand must actually be moving at 100mph as the ball is released... cool.

 

[Badly Shaved Monkey]

Hmmm... I suspect the answer is yes. I've seen some ultra-slow-mo footage of a golf ball being struck, and the ball definitely sprang away from the club head, even as the club head was still following-through. As you say, this seemed to be due to the stored energy in the deformed golf ball - it deformed quite significantly as the club head first hit it.

At the point of maximum deformation, the situation could be analysed like this - the club head and the golf ball are both moving at the same speed, but the golf ball is significantly deformed, thus providing a significant force in *both* directions - the "elastic force" from the ball is pushing the ball forward and the club back.

This isn't at all 'non-Newtonian', of course, but I'm sure you knew that

Thanks.

I did think that "non-Newtonian" in this context referred to objects that were not perfectly elastic when they impacted. Though it's 25 yrs since Applied Maths A-level.

 

[wollery]

Perfectly elastic means that all of the Kinetic Energy is conserved, but in any inelastic collision the momentum is conserved. Momentum is equal to mass times velocity (P=mv), so if the ball is light and the object striking it is heavy (or at least heavier than the ball), and follows through, then the ball will go off faster than the foot/club/bat that hit it was going.

This is totally Newtonian.

 

[Stimpson J. Cat]

Assuming that the ball is stationary, the actual maximum speed is twice the speed of the club. This speed is approached in the limit of the club being much more massive than the ball, and negligible energy being lost during impact.

The reason for this can be seen by looking at the impact in the reference frame of the club. There we have a ball hitting it as speed X. If the club is much more massive than the ball, it will not move significantly, so we basically have the ball bouncing off of the club. In the limit of negligible energy loss from deformation, the ball bounces back at the speed it hit at. Go back to the stationary frame, and we have the ball being hit off at twice the speed of the club. In reality, it will be somewhat less, because the club will slow down on impact, and the deformation of the ball (and club) will convert some energy to heat.

If the ball is not stationary (like in baseball) then it will be faster. Again, just look at things from the reference frame of the club (or bat), and it becomes pretty simple.


Dr. Stupid

 

[Badly Shaved Monkey]

Assuming that the ball is stationary, the actual maximum speed is twice the speed of the club. This speed is approached in the limit of the club being much more massive than the ball, and negligible energy being lost during impact.

The reason for this can be seen by looking at the impact in the reference frame of the club. There we have a ball hitting it as speed X. If the club is much more massive than the ball, it will not move significantly, so we basically have the ball bouncing off of the club. In the limit of negligible energy loss from deformation, the ball bounces back at the speed it hit at. Go back to the stationary frame, and we have the ball being hit off at twice the speed of the club. In reality, it will be somewhat less, because the club will slow down on impact, and the deformation of the ball (and club) will convert some energy to heat.

If the ball is not stationary (like in baseball) then it will be faster. Again, just look at things from the reference frame of the club (or bat), and it becomes pretty simple.


Dr. Stupid

Thanks