Range of Final Velocities in Collision of 10kg Ball and Sm. Mass Ball

[Hereformore]

Homework Statement

You have a 10kg ball that is going 10m/s. It is about to collide with a much less massive ball that is stationary.

What is the range of final velocities of the small mass?
All Collisions Are Elastic

Given Case 1: The large mass stops after the collision.

Given Case 2: The large mass keeps moving

Homework Equations

P=mv
Pi=Pf
KEi=KEf

The Attempt at a Solution

So formulaically, I understand why if the large mass stops after the collision the small mass could reach an infinitely high final velocity depending on how small it was. Or it could reach a velocity of Vf=Vi=10m/s if it were just a tiny bit less massive than the original mass (say 9.9999999kg).

What I don't understand is for the other case where the mass keeps moving.
If the large mass keeps moving, the small mass has to move at at least the speed of the large mass.
So if the smaller mass was just a tiny bit smaller than the large mass, then its final velocity could be Vi/2 = just about 5m/s (right?).

But if the smaller mass were infinitely small, I don't get why it's maximum velocity would be 2v. I Can run it through the formulas since momentum and kinetic energy is conserved, but intuitively i don't understand.

So A) Yes the smaler mass but be going at least as fast as the larger mass otherwise the larger mass would be "going through" the smaller mass

But how, if the smaller mass is infinitely small does it make sense that it could not go faster than 20 m/s or at 2vi? What relationship am i missing here?

 

[RMZ]

Hi,
You have some misconceptions,
First of all, if the mass of the smaller ball is negligible compared to the larger ball, it will take off with a velocity of at most twice that of he bigger mass.
Second, if the two masses are equal, then the stationary ball will take off with a velocity equal to that of the initially moving ball. In this case, the ball that was initially moving will completely stop upon contact with the second ball.
Therefore

So if the smaller mass was just a tiny bit smaller than the large mass, then its final velocity could be Vi/2 = just about 5m/s (right?).

is incorrect, because if it is just a tiny bit less massive than the first ball, it will take off at almost the exact initial speed of the first ball (while the first ball nearly stops).

Regarding your intuitive understanding of the concept, have you seen the derivation for the formula that backs these statements? Or does your book simply put the formulae on the page?

 

[mfb]

Given Case 1:[/B] The large mass stops after the collision.

Can you stop a moving massive ball by letting it collide with a table-tennis ball at rest?

If the large mass keeps moving, the small mass has to move at at least the speed of the large mass.

Is this really true? Think of billard and a second dimension.

So if the smaller mass was just a tiny bit smaller than the large mass, then its final velocity could be Vi/2 = just about 5m/s (right?).

Not in general.

But if the smaller mass were infinitely small, I don't get why it's maximum velocity would be 2v. I Can run it through the formulas since momentum and kinetic energy is conserved, but intuitively i don't understand.

Consider the frame of the heavy object. It acts like a wall then.

First of all, if the mass of the smaller ball is negligible compared to the larger ball, it will take off with a velocity of at most twice that of he bigger mass.
Second, if the two masses are equal, then the stationary ball will take off with a velocity equal to that of the initially moving ball.

It does not have to.

 

[RMZ]

I assumed a one dimensional elastic collision, in which case it does have to

 

[Hereformore]

Can you stop a moving massive ball by letting it collide with a table-tennis ball at rest?

Is this really true? Think of billard and a second dimension.

Not in general.

Consider the frame of the heavy object. It acts like a wall then.

It does not have to.

Thanks so much guys this is helping me clear up a lot of misconceptions.

So even under elastic collision conditions the larger mass would never be able to transfer all of it's kinetic energy to the second ball?

If both masses started moving, then wouldn't the smaller mass have to be at least the same speed as the large mass? Otherwise the large mass would "overtake" the small mass and go through it which would be impossible wouldn't it?

So if the two balls were very close in mass then the final velocity of the smaller object would be almost Vinitial and the velocity of the slightly larger object would be near zero because it trasnferred most of its kinetic energy to the smaller ball?

I'm having a bit of trouble understanding what you mean by frame of reference. The large ball approaching the small one (infinitely small mass), the large ball acts as a wall?

 

[RMZ]

1. No, otherwise in this case the larger ball would stop according to the work-energy theorem (W_{on ball}=deltaKl_{of that ball}.

2. The second ball will move with at least the same speed of the first, provided that the first strikes it. But if you try to move a very heavy object with a small one, much like a ball hitting a wall, then they won't move off with the same speed like youre saying (think about it, the first ball will bounce back while the giant one stayed almost still). That being said, for the case of a more massive object moving and striking a less massive object, the smaller mass will move off with at least the same speed as the bigger mass (like you said).3. Yes

4. He may have meant Center of mass reference frame. in this case, the much much larger object, which I believe he is considering a wall, appears to not really move while the smaller ball appears to move towards the 'wall'. So from this reference frame the ball appears to be moving towards the "wall", while the "wall" is almost stationary. So just imagine a rubber ball being thrown at a wall, there is no way it will shoot off at an infinitely high speed just because you decrease its mass or because the wall is made even more massive. Also, you can see more why the ball would only tke off with twice speed of the first (or in this reference frame, its initial speed). Tell me if you need me to explain the center of mass reference frame, it is not hard at all once you get it, and not understanding it may confuse you for 4.)'s explanation

 

[RMZ]

*For number four, he more likely meant to consider the reference frame of the heavy ball, as in, picture yourself viewing from a camera mounted on the heavy ball (you know what I am trying to say). The less massive ball that appeared stationary in the inertial reference frame will now seem to be approaching you at the speed the more massive object had in the inertial reference frame. so the ball appears to move towards this 'wall' with the more massive object's initial speed v while the massive object (or 'wall') appears stationary... and then follow the rest of my explanation for 4.) Sorry about that. either way I am saying pretty much the same thing, but this way is less complicated.

 

[mfb]

I assumed a one dimensional elastic collision, in which case it does have to

Who said the collisions would be one-dimensional? The question clearly needs two dimensions to be meaningful.

So even under elastic collision conditions the larger mass would never be able to transfer all of it's kinetic energy to the second ball?

Right.

If both masses started moving, then wouldn't the smaller mass have to be at least the same speed as the large mass? Otherwise the large mass would "overtake" the small mass and go through it which would be impossible wouldn't it?

Think about billard!

So if the two balls were very close in mass

They are not.

then the final velocity of the smaller object would be almost Vinitial and the velocity of the slightly larger object would be near zero because it trasnferred most of its kinetic energy to the smaller ball?

Again, think about billard.

I'm having a bit of trouble understanding what you mean by frame of reference. The large ball approaching the small one (infinitely small mass), the large ball acts as a wall?

Yes. As seen by the large ball, we have a tiny object approaching a resting really large mass - like a wall. The tiny object will keep its speed but reverse the direction...

 

[Hereformore]

Who said the collisions would be one-dimensional? The question clearly needs two dimensions to be meaningful.

Right.

Think about billard!

They are not. Again, think about billard.

Yes. As seen by the large ball, we have a tiny object approaching a resting really large mass - like a wall. The tiny object will keep its speed but reverse the direction...

I see. Great that makes a lot more sense now. Thanks for bearing with me! You guys are really saving me and making studying a lot more enjoyable haha.

One last question: in what situations would two balls rebound off each other? Is that only when both objects are moving towards each other?

 

[mfb]

What exactly do you count as rebound?

 

[Goldfox2112]

I have a similar question. How would you find the speed of a ping pong ball, going at -30 mph, on a head on collision course with a train, going at 60 mph, with the masses unknown, but the train being spectacularly massive compared to the ping pong ball.

 

[gneill]

I have a similar question. How would you find the speed of a ping pong ball, going at -30 mph, on a head on collision course with a train, going at 60 mph, with the masses unknown, but the train being spectacularly massive compared to the ping pong ball.

Please start a new thread of your own to ask your question. It is sufficiently different from the question posed in this thread to warrant a separate thread.