Deriving the Relative Velocity of a Ball Hit by a Massive Object

[imsmooth]

I was thinking about gravitational slingshots so I broke it down to a simple thought experiment: a ball being hit by a massive object. You have a ball of small mass m and a big moving object of large mass M moving at v = 10 m/s. The ball sees the wall approaching at 10 m/s. When the wall hits the ball the ball moves away from it at the same relative velocity, 10 m/s. Both frames should be the same if the ball moves to the wall or the wall move towards the ball. This means the ball has to be moving at 20 m/s in order to be moving away from the wall at the relative speed of 10 m/s. How does one derive this relationship mathematically.

 

[BvU]

he ball moves away from it at the same relative velocity, 10 m/s

Where did you find that ?
Important in collisions is momentum conservation. When collisions are elastic, you also have conservation of kinetic energy.

You can play with coins on a smoothly sliding table: big coin hits small coin ---> small coin moves off with much higher velocity.

Or think about a baseball player hitting a ball over the fence: would the slugger fly over the fence if he'd let it go ?

 

[imsmooth]

I want to add the ball for simplicity is at rest when hit.

 

[imsmooth]

Where did you find that ?
Important in collisions is momentum conservation. When collisions are elastic, you also have conservation of kinetic energy.

You can play with coins on a smoothly sliding table: big coin hits small coin ---> small coin moves off with much higher velocity.

Or think about a baseball player hitting a ball over the fence: would the slugger fly over the fence if he'd let it go ?

I just made this up in my head. Is there a mathematical way to show that the velocity after being hit is twice the velocity of what is hitting it (assuming the initial velocity of the object is zero).

 

[BvU]

Sure. Means its momentum is 0. As if you hit the ball from a pole. And the little coin was laying still (which was what I had in mind anyway).

Other example: golf ball from a tee !

 

[BvU]

Is there a mathematical way to show that the velocity after being hit is twice the velocity of what is hitting it

No there is not. Because it's not generally true !

Thought experiments should start from true statements ... or axioms that can't be proven wrong (no expert in this area, so someone please correct me there if I'm wrong)

 

[PeroK]

I was thinking about gravitational slingshots so I broke it down to a simple thought experiment: a ball being hit by a massive object. You have a ball of small mass m and a big moving object of large mass M moving at v = 10 m/s. The ball sees the wall approaching at 10 m/s. When the wall hits the ball the ball moves away from it at the same relative velocity, 10 m/s. Both frames should be the same if the ball moves to the wall or the wall move towards the ball. This means the ball has to be moving at 20 m/s in order to be moving away from the wall at the relative speed of 10 m/s. How does one derive this relationship mathematically.

You got the right answer, although I would have thought about it from the frame of the large object: the small object moves towards it at 10m/s, collides elastically (which means the ball bounces off at 10m/s). Hence, in the ball's frame it's now moving at 20m/s. Assuming the wall is too large to be affected.

How much physics and maths do you know? You can study this collision and if you take the mass of the wall to be very large compared to the mass of the ball, then you will get the above result (as a good approximation).

 

[imsmooth]

So does the ball move away from the wall at twice the wall's approach speed?

 

[PeroK]

So does the ball move away from the wall at twice the wall's approach speed?

Yes! Approximately.

 

[BvU]

PeroK: thanks for correcting me ! Bad moment on my part , sorry I'mS !

Defence: some of the confusion from the transition from "moving massive object" to "non-moving" wall, to be sure !

 

[PeroK]

PeroK: thanks for correcting me ! Bad moment on my part , sorry I'mS !

I've had a few bad moments this year as well!

 

[imsmooth]

lets say the ball is 1kg, v = 0 and the wall is 1000kg, v = 10 m/s

1kg*0 + 1000kg*10 m/s = 1kg(v1) + 1000kg* (v2)

I don't know what the reduction in the wall's velocity. Do I need to also consider the kinetic energy?

 

[PeroK]

lets say the ball is 1kg, v = 0 and the wall is 1000kg, v = 10 m/s

1kg*0 + 1000kg*10 m/s = 1kg(v1) + 1000kg* (v2)

I don't know what the reduction in the wall's velocity. Do I need to also consider the kinetic energy?

Yes, although I suggest you imagine the ball is hitting the wall. I.e. from the wall's frame. It might be easier to see what happens that way. Both ways will work though.

 

[A.T.]

How does one derive this relationship mathematically.

https://en.wikipedia.org/wiki/Elastic_collision#One-dimensional_Newtonian

 

[Mark Eaton]

The equal action would equal the reaction or visa versa. But the ball, will be reacting to the force from hitting the wall. The force applied to he ball will be a action, not a reaction. Unless the ball is going to move the wall and apply a force to it., when it hits it, as apposed to the wall causing reaction to the ball, if the wall is not moving
So the ball must be moving at twice the velocity to cause this reaction., as apposed to the action.

 

[imsmooth]

I worked through the mass last night and got the correct derivation. I bit of simple, but tedious algebra; not a big problem.