Achieving a step function with velocity

Question

I need some help clarifying step functions. My question is can velocity step up? For example, is it possible for a body to accelerate from 0 to, say, 60 miles per hour in an infinitesimal or size-zero interval of time?

Above image just to provide context

If the answer is no, I want to know why is that? My reason for asking is that whenever a body begins to accelerate from a velocity 0 to any final velocity V, then the instant it is no longer stationary, it has a velocity v. This velocity v would, in most observed cases as I know them would be very small, but if we change some parameters like decreasing the mass and/or increasing the force, can't we increase the velocity v to make it approach or reach velocity V, hence achieving the step function?

Thanks!

Answer 1

A true step function in velocity would mean infinite acceleration which would require an infinite force. It is not possible. It can be an excellent approximation if the time period of the acceleration is very short compared to other time constants in the problem. If you think of a marble bouncing on a hard floor, the time of acceleration is very short compared to the time the marble is falling before and rising after the impact. It is not a bad approximation to say the marble and floor are rigid, the impact is instantaneous, and compute the behavior that way. Someone with a high speed camera could capture some frames showing the marble and floor deforming and measure the time of impact and hence the time of the acceleration. This shows the velocity is not a true step function, but it may be close enough to one for the purpose at hand.

Answer 2

A model that you may often encounter is that at time $t_0 = 0,$ an acceleration is applied to the object that causes its velocity to steadily increase from $0$ to some positive value. If at time $t_1 > 0$ you find that the object was moving at velocity $v_1,$ then at time $t_1/2$ it was moving at velocity $v_1/2.$ No matter how close $t_1$ is to zero and no matter how small $v_1$ is, there is always an earlier time after the start of motion and always a smaller velocity.

Even this model is somewhat suspect because it assumes the acceleration goes from nothing to its full value instantly. This does not generally happen in the real world. But it is often a reasonable approximation of what happens.

In the same way, an instantaneous jump from one speed to another may also be a reasonable approximation, as already noted in another answer. But there is no compelling reason to make this approximation other than that it might be convenient and good enough.