- #1

vbrasic

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## Homework Statement

A proton is initially located at the origin of some coordinate system (at rest), when a time-dependent force, $$F(t)=F_0\sin{(\omega t)},$$ is applied to it, where ##F_0## and ##\omega## are constants.

a) Find the velocity and displacement of the proton as functions of time.

b) Show that the proton's distance from the origin grows without bound as time goes on.

## Homework Equations

##F(t)=ma(t)## seems to be the only necessary equation.

## The Attempt at a Solution

First, we solve for ##a(t)## using ##F(t)=ma(t)##. We have $$ma(t)=F_0\sin{(\omega t)}\rightarrow \frac{dv}{dt}=\frac{F_0}{m}\sin{(\omega t)}.$$ Then we integrate once to find velocity of the particle. We have, $$\int_{0}^{v}dv'=\int_{0}^{t}\frac{F_0}{m}\sin{(\omega t)}\,dt'\rightarrow v(t)=-\frac{F_0}{m\omega}\cos{(\omega t)}.$$ Integrating a second time, we find that the proton's displacement is given by, $$x(t)=-\frac{F_0}{m\omega^2}\sin{(\omega t)}.$$ I'm having trouble showing that the proton's distance from the origin grows without bound, as it seems that the proton should just oscillate between, ##\frac{F_0}{m\omega^2},##, and, ##-\frac{F_0}{m\omega^2}##.

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