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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)

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    $\begingroup$ How do you know that acceleration is not constant? Are there dissipative forces that aren't mentioned here? $\endgroup$ – probably_someone Jul 12 '18 at 19:44
  • $\begingroup$ Standard equations for distance in newtonian mechanics can be derrived by integrating vdt and/or tdv as appropriate. dv is caused by the engine. Or you could just look them up. $\endgroup$ – JMLCarter Jul 12 '18 at 19:53
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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 $P=Fv$.

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