# I How is initial acceleration accounted for in the impact of two bodies

#### timza

Imagine a moving rigid body B1 like a golf clubhead impacts with a stationary elastic body B2 like a golfball. Imagine that the initial velocity of the golfball at time T1 is zero, and the final velocity of the golf ball at time T2 is some value. Imagine that time T1 is when the clubhead first strikes the elastic golfball, and time T2 is when the elastic golf ball has retained its round shape and leaves the clubhead. Imagine we can know what the moment of inertia and angular velocity is of the golfer/shaft/clubhead system B1 at T1. I realize that I just kind of threw in a wrench when I then spoke of B1 as not just being a clubhead, but instead now a system of golfer/shaft/clubhead. But the face of the clubhead is rigid.

Question: For three scenarios, where the clubhead velocity at T1 is exactly the same for each scenario, is the resultant final velocity of the golfball B2 greater, the same, or lesser, if at impact time T1 the clubhead B1 is: 1. moving at a constant velocity, 2. increasing in velocity (accelerating), or 3. decreasing in velocity (decelerating)?

If I knew the moment of inertia and the angular velocity of the golfer/shaft/clubhead B1, at impact time T1 I would know the initial momentum of the collision because the golfball B2 is at rest and has zero momentum. So then if I knew the mass of the golf ball and assumed flush contact and linear momentum, and no golfball spin after impact, I could use the conservation of momentum to find the final golfball velocity at time T2. But I think that the conservation of momentum is based on both bodies under constant speed.

If I wanted to calculate the force during the deformation of the elastic ball I could video the impact and measure the time between the start of the impact at time T1 to the release of the ball at time T2, and impulse equals force times delta time equals mass times change in velocity. I have my change in golfball velocities from the above conservation of momentum, and now I have the change in time, so I could find my force.

But, if the clubhead was accelerating at the start of impact at time T1, I would say that there was/is a force that was/is continuing to act on that shaft/clubhead system through impact. So back to the impulse equation. Now I could assume that since the golfer/shaft/clubhead is accelerating at T1 that the force that causes the accelerating continues through impact, and add that force to the impulse equation, to find a new, greater final velocity of the ball at time T2.

Whew. Where am I wrong? Is there a different equation that I don't even know enough about to Google for that would involve Initial and Final accelerations during impact, or involve an additional constant force through impact? I do not need to calculate actual numbers. I just want to study and learn the concepts.

I have for years talked to golfers who have just enough knowledge of physics to believe that golfball impact is only about the velocity of the clubhead and the force to the golf ball. They live in an F = ma world and often believe that if you cut the clubhead from the system of the golfer/shaft right before impact the distance the ball would go would be the same as if the clubhead remained connected to the golfer/shaft. They do not agree with me that you can increase the moment of inertia of the golfer/shaft/clubhead system to increase final golfball velocity after impact, even at the same measured clubhead velocity. But now I am thinking that I have been missing something in the force that the golfer continues to apply to the shaft/clubhead at and through impact.

Thank you. First time post. My searching led me to this forum. But I did not find anything specific on these concepts.

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#### BvU

Homework Helper
Hello timza, :welome:
For three scenarios
My ten cents is that an accelerating club head can transfer more momentum, because, thanks to its increasing velocity it will remain in contact with the ball a little longer.
But I think that the conservation of momentum is based on both bodies under constant speed
Conservation of momentum is based on Newton 3 : the force the club exercises on the ball is equal to the force the ball exercises on the club. And with $F = {d\over dt} mv$ the sum of opposing forces being zero results in the sum of momenta being constant.
But, if the clubhead was accelerating at the start of impact at time T1, I would say that there was/is a force that was/is continuing to act on that shaft/clubhead system through impact. So back to the impulse equation. Now I could assume that since the golfer/shaft/clubhead is accelerating at T1 that the force that causes the accelerating continues through impact, and add that force to the impulse equation, to find a new, greater final velocity of the ball at time T2.
I agree -- see previous paragraph (but perhaps my betters can put us right)
Whew. Where am I wrong?
Says who ?

There are some impressivve videos here and here
In this video at 1:30 you can see the club bend like a reed -- the head is definitely accelerating

#### timza

In video, the golf shaft bending toward the ball as the clubhead reaches the ball, is called the jello effect, and is due to the image being taken line by line from top to bottom, so further down the image, what is shown lower on one frame actually happened later than what is recorded higher on the image. If you were to take a correct picture in the dark with a flash at impact of an experienced golfer the shaft would bow back not forward. I do not want to get off on a rabbit trail on this point. I would be willing to get on a rabbit trail of flippers vs swingers vs hitters though. It is my belief that hitters in golf and baseball continue to apply a force to the handle with the straightening and thrusting of their rear elbow (right elbow for a right handed golfer), and therefore have a greater moment of inertia of the system of golfer and shaft and clubhead at impact, because the system is more connected at impact.

#### BvU

Homework Helper
You're absolutely right - my apologies: it wasn't a high-speed recording like the other two at all .
However, 'bowing back' does not mean 'not tangentially accelerating', so that can argument can stay standing -- I suppose.

#### timza

Yes. If a golfer was purely pulling the club, along the shaft of the club, I could imagine the club not bowing, because there would not be any force being applied to the handle with the rear elbow thrust. But it could still be accelerating due to the hands pulling along the shaft of the club, and gravity pulling it down. Just not bowing.