Momentum

[chiddler]

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A 2 kg ball moving at 4 m/s collides elastically with a 6 kg ball that is stationary. The 2 kg bounces off at 2 m/s. What is the speed of the 6 kg ball?

Answer: 2 m/s.

The solution used initial = final kinetic energy, 1/2mv^2, instead of momentum, mv. Why can't I use mv? You get different answers:

(2)(4) + 0 = 2(2) + 6*Vf
8 = 4 + 6Vf
4 / 6 = 2 / 3 = Vf

Thanks.

 

[Pisiform]

(2)(4) + 0 = 2(-2) + 6*Vf
8 = - 4 + 6Vf
vf = 12/6 = 2

you need to take care of the sign.If you take right as positive then left would (rebounce) would be negative)

 

[chiddler]

(2)(4) + 0 = 2(-2) + 6*Vf
8 = - 4 + 6Vf
vf = 12/6 = 2

you need to take care of the sign.If you take right as positive then left would (rebounce) would be negative)

that never occurred to me, thanks.

 

[chiddler]

Another question, please:

If a lighter ball collides with a heavier one, why does it bounce back? And a larger ball --> smaller one when both move in the same direction. Why?

 

[MT Headed]

Another question, please:

If a lighter ball collides with a heavier one, why does it bounce back? And a larger ball --> smaller one when both move in the same direction. Why?

The best way to answer these types of questions on the mcat is to think in extremes.

What if a ping pong ball hits the side of a building. Why does it bounce back?

What if a semi truck hits a piece of sand on its windshield as it cruises down the highway? Why do the truck and dust now go in the same direction?


If these extremes don't do it for you, it can also help to change your frame of reference. I find students have a harder time with your second question, so try to think about it from the frame of reference of somebody inside the truck. As far as they can tell, from their perspective, the truck is sitting still. Suddenly a speck of sand slams into the front of the truck, at 60mph. Sure it is going to bounce off of the truck and go 60mph in the opposite direction, and the truck won't even notice.

Now add 60mph of forward direction to each object. The truck is always going at 60mph. The sand's initial velocity is 0mph, perfectly still. The final velocity of the sand? 120mph, in the same direction as the truck.

 

[milski]

Another question, please:

If a lighter ball collides with a heavier one, why does it bounce back? And a larger ball --> smaller one when both move in the same direction. Why?

If your moving ball has mass M, the stationary one m, the initial speed is v and after the collision their speeds are v1 and v2, positive direction is to the right:

Mv=Mv1+Mv2 // from conservation of momentum
1/2*(Mv^2)=1/2*(Mv1^2+mv2^2) // from conservation of energy

When you solve these two for v1 and v2:

v1=v(M-m)/(m+M)
v2=2vM/(m+M)

That tells you that the stationary ball always starts moving in the direction of the initial motion.
You have three cases for the moving ball:
- if it has the same mass, M=m, the moving ball stops, v1=0
- if the moving ball is heavier, M>m, v1>0, it continues to move in the same direction it was
- if it was ligher, M<m, v1<0, it bounces back

 

[chiddler]

The best way to answer these types of questions on the mcat is to think in extremes.

What if a ping pong ball hits the side of a building. Why does it bounce back?

What if a semi truck hits a piece of sand on its windshield as it cruises down the highway? Why do the truck and dust now go in the same direction?


If these extremes don't do it for you, it can also help to change your frame of reference. I find students have a harder time with your second question, so try to think about it from the frame of reference of somebody inside the truck. As far as they can tell, from their perspective, the truck is sitting still. Suddenly a speck of sand slams into the front of the truck, at 60mph. Sure it is going to bounce off of the truck and go 60mph in the opposite direction, and the truck won't even notice.

Now add 60mph of forward direction to each object. The truck is always going at 60mph. The sand's initial velocity is 0mph, perfectly still. The final velocity of the sand? 120mph, in the same direction as the truck.

When you say add 60 mph of forward direction to each object, sand and truck, then they will not collide because both moving same speed same direction. Am I misunderstanding or is that a mistake?

I know, from analyzing milski's equations, that a truck colliding with a stationary grain will cause the grain to move at 2*velocity of the truck (and you wrote 120 mph in the same direction). Can you please explain why this occurs?

If your moving ball has mass M, the stationary one m, the initial speed is v and after the collision their speeds are v1 and v2, positive direction is to the right:

Mv=Mv1+Mv2 // from conservation of momentum
1/2*(Mv^2)=1/2*(Mv1^2+mv2^2) // from conservation of energy

When you solve these two for v1 and v2:

v1=v(M-m)/(m+M)
v2=2vM/(m+M)

That tells you that the stationary ball always starts moving in the direction of the initial motion.
You have three cases for the moving ball:
- if it has the same mass, M=m, the moving ball stops, v1=0
- if the moving ball is heavier, M>m, v1>0, it continues to move in the same direction it was
- if it was ligher, M<m, v1<0, it bounces back

Thanks this was very helpful.

 

[milski]

When you say add 60 mph of forward direction to each object, sand and truck, then they will not collide because both moving same speed same direction. Am I misunderstanding or is that a mistake?

Initial experiment is truck at 0 km/h, object -60 mph, bounces at 60 mph.
In the new reference frame, where you add 60 mph, truck is 60 mph, object at 0 mph, bounces at 120 mph.

I know, from analyzing milski's equations, that a truck colliding with a stationary grain will cause the grain to move at 2*velocity of the truck (and you wrote 120 mph in the same direction). Can you please explain why this occurs?

Thanks this was very helpful.

Think of it as a ball bouncing against a wall. The ball is headed towards the wall, hits the wall, the wall does not care, so the ball heads back with all of its energy, which makes it go in the opposite direction with the same speed. (Very much like a scene from Star Wars where Han Sole chases a startrooper and soon comes back at the same speed and opposite direction, followed by a squad of startroopers). Based on the frame of reference, the ball/grain speeds can be either -60/60 or 0/120 - it is the same thing.

 

[chiddler]

Initial experiment is truck at 0 km/h, object -60 mph, bounces at 60 mph.
In the new reference frame, where you add 60 mph, truck is 60 mph, object at 0 mph, bounces at 120 mph.


Think of it as a ball bouncing against a wall. The ball is headed towards the wall, hits the wall, the wall does not care, so the ball heads back with all of its energy, which makes it go in the opposite direction with the same speed. (Very much like a scene from Star Wars where Han Sole chases a startrooper and soon comes back at the same speed and opposite direction, followed by a squad of startroopers). Based on the frame of reference, the ball/grain speeds can be either -60/60 or 0/120 - it is the same thing.

oh damn! you're right! i never realized this o_o

if grain is at rest and is hit by truck, why doesn't it go just as fast as the truck? I understand what you are saying about the two situations being the same, but trying to illustrate this scene realistically is giving me trouble.

 

[MT Headed]

Have you ever played or seen tee-ball? The baseball sits on a stand, and the little kid whacks it. The light ball takes off much faster than the heavy bat.

Or how about golf, where the light golf ball takes off much faster than the heavy clubhead that was swung.

I know this is a hard concept to grasp, and sometimes it is just easier to think of the perspective of the heavy object. From that point of view, something light and insignificant arrives, and bounces off just as quickly.

Sometimes you just need to sleep on it for a day or two 🙂

 

[milski]

oh damn! you're right! i never realized this o_o

if grain is at rest and is hit by truck, why doesn't it go just as fast as the truck? I understand what you are saying about the two situations being the same, but trying to illustrate this scene realistically is giving me trouble.

Going as fast as the truck is the same as having the truck stationary and the ball just sticking there without bouncing back. From a physical point of view, that would be the equivalent of transferring all of the KE of the grain to the truck. A bug splashing on the windshield would be a good example of that.

Another way to look at it: think about being in a train and throwing a ball (the grain) against the back wall of the train (the truck). Consider throwing the ball back as fast as the train is moving. To someone sitting with you, it's fairly obvious that the ball will bounce back. For someone looking from outside, once you throw the ball, it will be stationary (v-v). Once it hits the wall, it will start moving in the opposite direction twice as fast as the train. (v+v)

 

[chiddler]

Sometimes you just need to sleep on it for a day or two 🙂

amazing how well this works sometimes!

Going as fast as the truck is the same as having the truck stationary and the ball just sticking there without bouncing back. From a physical point of view, that would be the equivalent of transferring all of the KE of the grain to the truck. A bug splashing on the windshield would be a good example of that.

Another way to look at it: think about being in a train and throwing a ball (the grain) against the back wall of the train (the truck). Consider throwing the ball back as fast as the train is moving. To someone sitting with you, it's fairly obvious that the ball will bounce back. For someone looking from outside, once you throw the ball, it will be stationary (v-v). Once it hits the wall, it will start moving in the opposite direction twice as fast as the train. (v+v)

ok this makes a lot more sense now.

Thank you both for the help. Is very much appreciated.