Calculate the change in velocity over a given distance.

Question

I am trying to calculate the maximum change in speed over a given distance with a known constant acceleration, current speed and of course, distance. I do not know the time.

However, I run into the issue that the speed is undefined if the starting speed is zero.

Given a=∆v/t, ∆v=a*t, and ∆t=d/v, I came up with ∆v=a*(d/v). However if the current speed is zero, I get a undefined value for the delta speed.

I feel as if I am missing something obvious, or that the algorithms I am using are simplified versions of more complete ones.

Thanks in advance for any help.

Answer 1

If we measure from the starting location and the acceleration is constant, the position as a function of time is $$s(t)=v_0t+\frac 12at^2$$ where $v_0$ is the starting velocity and $a$ is the acceleration. We can find the time to get to position $d$ by solving the quadratic $$d=v_0t+\frac 12at^2\\ t=\frac 1{a}\left(-v_0+\sqrt{v_0^2+2ad}\right)$$ Then the change in speed is just $at$

Answer 2

Use conservation of energy to eliminate those unknown times entirely. The change in kinetic energy is equal to $\frac12mv_{\text{final}}^2-\frac12mv_{\text{init}}^2$. The work done in accelerating the object is $mad$ (using the convention that this will be negative for deceleration). Equate these and solve for $v_{\text{final}}$: $$v_{\text{final}}=\sqrt{v_{\text{init}}^2+2ad}.$$ Note that if there’s a change in direction as a result of this acceleration, you’ll need to split the distance $d$ into two segments.