Distance traveled by accelerating vehicle

Question

Let's say I'm trying to model a train accelerating from speed $v_1$ to speed $v_2$, where the train's mass $m$ and engine power $P$ is known. I would like to find both the time $t$ and the distance $s$ needed to reach speed $v_2$. I know that the work done by the engine, $A=Pt$, would be equal to the change in kinetic energy $∆E$. So:

$$ \frac{1}{2}m(v_2^2-v_1^2)=Pt $$ $$ t=\frac{m}{2P}(v_2^2-v_1^2) $$ But how would I go about finding the distance traveled during time $t$, because acceleration is not constant? I'm not well versed in calculus, but maybe it has something to do with integrals? (distance being the integral of velocity)

Answer 1

One can use the fact that the kinetic energy increases linearly in time:

$\frac{1}{2}mv^2(t)=\frac{1}{2}mv_{0}^2+Pt$

to find velocity as a function of time itself: $v(t)=\sqrt{v_0^2+\frac{2Pt}{m}}$

Now, noticing that $v=\frac{dx}{dt}$, you can obtain the answer by integrating in time once (I'll leave this elementary integral to the reader). The acceleration is definitely not constant, since the velocity is sublinear in t. An equation in terms of time for the magnitude of the force can be found by using the relation $P=Fv$.