Velocity change when an impulsive force is applied ?

[shalikadm]

I was reading an article about velocity and displacement of a particle..It says "though the position of a particle is a continuous function of time,velocity sometimes isn't.For instance velocity is not continuous while an impulsive force is being applied."
But I think that velocity changes continuously even an impulsive force is being applied.imagine a ball is collided into a bat.when it collides its velocity is reduced to zero and increased into some different velocity but continuously..Please explain this to me..

 

[NascentOxygen]

It may simply be a matter of semantics that you are wrestling with. Imagine a steel ball rolling to the right at 2 m/sec. when at t=5 sec it collides with an identical ball moving to the left at the same speed. A graph of velocity vs. time would show a horizontal line of, say, +2m/sec before t=5 sec, and after that it (on our scale) "immediately" switches over to a horizontal line of -2m/sec. This does appear to me, for all practical purposes, to be a discontinuity.

Of course, with steel being a relatively elastic material, during the interaction, each ball does on a microscopic level steadily slow, then steadily speed up. It has to, because a perfect discontinuity in the speed would correspond to infinite acceleration meaning an infinite force.

So, depending on perspective or the context, you could say velocity is continuous — or it isn't. Both statements are right, or close enough to being right that the difference doesn't matter. That's my view.

 

[AlephZero]

Mathematically, you can think of an finite impulse (change or momentum) as the limiting case of a very large force acting for a very short time, as the force becomes "infinitely large" and the time "infinitely small" (i.e. zero). In that sense there is a step change in the velocity of the particle.

This idea of an impulse acting "instantaneously" is not physically correct, but if the duration of the impulse is very small compared with the time scale you are interested in, it simplifies the math. For example in collisions between objects, often you are not interested in what happens to the objects DURING the collision, but only in their velocities AFTER the collision.

I think your quote would be better English and less confusing if it said "For instance velocity is not continuous WHEN an impulsive force is being applied", not WHILE.

 

[shalikadm]

It may simply be a matter of semantics that you are wrestling with. Imagine a steel ball rolling to the right at 2 m/sec. when at t=5 sec it collides with an identical ball moving to the left at the same speed. A graph of velocity vs. time would show a horizontal line of, say, +2m/sec before t=5 sec, and after that it (on our scale) "immediately" switches over to a horizontal line of -2m/sec. This does appear to me, for all practical purposes, to be a discontinuity.

Of course, with steel being a relatively elastic material, during the interaction, each ball does on a microscopic level steadily slow, then steadily speed up. It has to, because a perfect discontinuity in the speed would correspond to infinite acceleration meaning an infinite force.

So, depending on perspective or the context, you could say velocity is continuous — or it isn't. Both statements are right, or close enough to being right that the difference doesn't matter. That's my view.

I think that the impulsive force generated by one ball on the other slows down to o ms^-1 till it changes all its kinetic energy into elastic potential energy and speeds up to 2 ms^-1 in the other direction changing all elastic potential energy into kinetic energy in milliseconds.I also thinks that there's an impulsive acceleration generated by this impulsive force which changes the velocity continuously..like velocity increment in free fall-starting at o ms^-1 and increasing velocity continuously like 0.00000000001 ms^-1 in 0.0000000000001 seconds..

 

[shalikadm]

Mathematically, you can think of an finite impulse (change or momentum) as the limiting case of a very large force acting for a very short time, as the force becomes "infinitely large" and the time "infinitely small" (i.e. zero). In that sense there is a step change in the velocity of the particle.

This idea of an impulse acting "instantaneously" is not physically correct, but if the duration of the impulse is very small compared with the time scale you are interested in, it simplifies the math. For example in collisions between objects, often you are not interested in what happens to the objects DURING the collision, but only in their velocities AFTER the collision.

I think your quote would be better English and less confusing if it said "For instance velocity is not continuous WHEN an impulsive force is being applied", not WHILE.

Is it something like this ?
-we just take velocity changing discontinuously mathematically though it changes continuously.Do we just dismiss what happens actually inside it? So how can we say that the velocity changes discontinuously ?

 

[xAxis]

Yes. We just dismiss what happens during that small time period while the balls are in contact. We model that time to be zero, so that, for instance, if the collision happens at t = t1, then for any other t > t1, the speed of ball is taken to be v(t+dt), where dt is the actual time of collision.

 

[Chestermiller]

Please see my responses in the following thread, particularly response # 22:

https://www.physicsforums.com/showthread.php?t=649233

The key to understanding what is going on in impulsive collisions like this is to recognize that the deformation of each of the objects is non-homogeneous and varies with time, as elastic compression waves travel from the contact end of each object to the free end, and then the compression is released from the free end to the contact end. In my responses, I also present the analytic results for two identical elastic cylinders colliding head on.