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I think if $F(t) = kt^0$ then $$x(t) = x_0 + v_0t + \frac{k}{m}\frac{t^2}{2!},$$ and if $F(t) = kt^1$ then $$x(t) = x_0 + v_0t + \frac{k}{m} \frac{t^2}{2!} + \frac{k}{m} \frac{t^3}{3!},$$ and so on, is this correct? I do not know how to write it better, maybe like this but im not sure.

If $F(t) = kt^n$ then $x(t) = x_0 + v_0t + \sum_{i=2}^{n+2}({\frac{k \cdot t^i}{m\cdot i!}})$

I found this other question of which I think the answer is suspiciously similar Kinematic equation as infinite sum but I can't completely understand the answer since I'm not good with derivatives.

Qmechanic
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clara raquel
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1 Answers1

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Your second equation is incorrect. Given Newton's second law $F=m\ddot{x}$, we see that on the left side we have $F=kt$ and on the right side $m\frac{d^2}{dt^2}(x_0+v_0t+\frac{k}{m}\frac{t^2}{2!}+\frac{k}{m}\frac{t^3}{3!})=k+kt$, such that $F\neq m\ddot{x}$ for any $k\neq 0$.

Given a force $F=kt^n$, the expression for $x$ is quite simple:

$$m\ddot{x}=kt^n\\\ddot{x}=\frac{k}{m}t^n \\ \dot{x}=\int\frac{k}{m}t^n dt=\frac{k}{m}\frac{t^{n+1}}{n+1}+v_0 \\ x=\int\left(\frac{k}{m}\frac{t^{n+1}}{n+1}+v_0\right)dt=\frac{k}{m}\frac{t^{n+2}}{(n+2)(n+1)}+v_0t+x_0$$

Sturrum
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