# Does the ball stop?

#### Pip Threlfall

When I was at school 50 years ago we had this problem to solve in physics. A steel ball is dropped onto a steel plate, calulate how high it will bounce.

I said that its initial volocity on the bounce would be zero. The teacher said it would be the same as the final volocity when it hit the steel plate. I said it stopped, he said it only changed its direction. I just cannot see how it cannot stop even if it is for no time. Anyway the upshot was he thought me incapable of grasping the basic physics. Comments please.

#### arbolis

PHF Hall of Fame
Hi and welcome here.
A steel ball is dropped onto a steel plate, calulate how high it will bounce.
It is impossible to know, unless we're given the coefficient of restitution. If the collision is elastic then the ball will bounce as high as it has been initially dropped. In reality there is no such case as an elastic collision so one must deal with the coefficient of restitution.

I said that its initial volocity on the bounce would be zero. The teacher said it would be the same as the final volocity when it hit the steel plate. I said it stopped, he said it only changed its direction. I just cannot see how it cannot stop even if it is for no time. Anyway the upshot was he thought me incapable of grasping the basic physics. Comments please.
Of course you are right to say that at a given time the velocity of the ball is the null vector. (0m/s). This is when the ball touch the plate.
Think about this : the velocity of any body is a continuous function. So passing from negative values to positive values without passing by the null value is impossible.

#### physicsquest

PHF Helper
Yes it does go to zero. The kinetic energy is converted into P.E. because of deformation of the ball and the surface (though it may be small) and then reconverted back to K.E.
Think of it falling on a spring which is not very stiff instead. The spring will be compressed and the vel will become zero.Then the spring will hurl the ball back upward to the same height.(assuming no loss of energy)
Now think of the spring getting stiffer and stiffer; the compression will also be progresively less. However the ball will rise to the same height. Our scenario is quite similar.
Conversely think of the ball being spongy. It will rise to a very low height, but the vel does go to zero.

#### Parvez

PHF Hall of Honor
This is very important to consider at what standard this was taught to you. You said you were in school. At that level it was not appropriate for a teacher
to discus the minute details of the changes taking place during the collision.
Further the initial velocity after the collision does depends upon how you define it. The falling object stops for a while during collision but for the purpose of using the equation of motions (v = u + at etc) the initial velocity is defined as the velocity obtained by the object when the potential energy of the object due to deformation etc becomes zero or in simple words it is the velocity just after the object leaves the surface.

If what you say is correct then the final velocity of the falling object will also be zero. But then it again depends how you define the final velocity. If you want to use equations of motion then the final velocity is the velocity just before the object hits the surface.

And finally if the two surfaces are perfectly elastic(theoretically) they will take no time to recover their original shape after deformation and in that case this is only the change in direction without stopping.
I have seen students who do not listen to the teachers and even before the teacher begins to speak they have there own agenda in mind to contradict what the teacher says. They thus blocks the inflow of the information and remains confused for ever.

So I think the teacher taught to you what he considered correct at that level. But "you learn to unlearn again" - says Feynman

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#### physicsquest

PHF Helper
And finally if the two surfaces are perfectly elastic(theoretically) they will take no time to recover their original shape after deformation and in that case this is only the change in direction without stopping.

Could you explain the bold part?

Think about this : the velocity of any body is a continuous function. So passing from negative values to positive values without passing by the null value is impossible.

How would you explainthis otherwise?

#### Pip Threlfall

To "recover their original shape" change must take place, but change cannot happen without time because time is change of position of particals in space. So there still must be an instant where deformation has ceased and recovery has not yet started. Could this be the Singularity that Science has been searching for?

#### physicsquest

PHF Helper
We have to be careful when we use the word singularity. It normally
refers to a case where a fuction blows up say because the denominator goes to zero at a point . Here the case doen't seem that drastic as it is the displacement vector which reverses during a very small time interval and hence i dont think this is the singularity science has been looking for.

arbolis

#### arbolis

PHF Hall of Fame
To "recover their original shape" change must take place, but change cannot happen without time because time is change of position of particals in space. So there still must be an instant where deformation has ceased and recovery has not yet started. Could this be the Singularity that Science has been searching for?

Absolutely not!!!! Time is not dependent of motion (which is the change of position with respect to time). Time pass even if your object is at rest! As I said, there will be a time when the ball has no velocity and it is exactly when the ball touches the plate (this is right if we simplify the problem so that we consider no deformation of the ball and plate during the collision. If we do not simplify the problem like this, there will still be a moment where the velocity of the ball is 0m/s but it would be harder to determine exactly when it occurs)

Think about this : the velocity of any body is a continuous function. So passing from negative values to positive values without passing by the null value is impossible.

How would you explainthis otherwise?
What I said is very important, it clearly implies that the ball has to have a zero velocity vector if it goes downward and then upward.
I'm not sure I could say it in other words. Ok... the velocity cannot jump from positive values to negative ones without passing by 0m/s, otherwise the object would not have a position for a moment, which is absurd.

Edit : Let me add that there is an impulse between the ball and the plate when they are in contact, and the speed goes down very quickly to be 0 m/s for a moment and then increases. Does it make more sense?

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physicsquest

#### Parvez

PHF Hall of Honor
And finally if the two surfaces are perfectly elastic(theoretically) they will take no time to recover their original shape after deformation and in that case this is only the change in direction without stopping.

Could you explain the bold part?

How would you explainthis otherwise?
Well, I described the situation wrong. The correct thing would be to say "If the collision is perfectly elastic the body would not deform at all and will take no time to rebound.

Think about this : the velocity of any body is a continuous function. So passing from negative values to positive values without passing by the null value is impossible.

Mere passing through a magnitude of zero velocity does not mean that the body has stopped. Think of the definition of a stationary body - if it does not change its position with time wrt to certain frame of reference. Now if at an instant the body has certain position, what is position next instant ? If it is still there then it has stopped but if at next instant its position is different from the previous one, it has not stopped.
Consider the motion of upward thrown body. Its velocity decreases becomes zero at top most point and again increases. It has not stopped there though it passed through zero velocity. I think you must be certainly aware of the question teachers often asks to high school students - "can a body have zero velocity but still accelerating?" and told the answer that this happens at the top point of the upward thrown object. Now if it has stopped at that point, how can it be accelerated.

Edit : I think a stretched it too far. This all may plase be ignored.

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arbolis

#### physicsquest

PHF Helper
Stretching leads to increase in P.E. and is also good exercise!

Doesn't it have something to do with limits?

For example, vel = delta x / delta t , yet we talk of the velocity at an instant t! though actually it is lim [(t + delta t)- t] as delta t tends to zero.

Parvez