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This is my first time in Physics forums so forgive me if I stray from the convention for asking questions. Me and a friend are completely stumped on our first assignment for the year and we are not sure where else to turn!

1. The problem statement, all variables and given/known data

A particle of mass m can move in a straight line. The only force acting on the particle opposes its motion, and depends only on its speed. The time T taken for the particle to come to rest after it is set into motion with speed u is given by the formula [itex]T=klog(1+au)[/itex] for all u, where k and a are positive constants. Find the force acting on the particle during the interval [itex]0\leq t\leq T[/itex] as a function of its speed v. Find also the distance the particle travels in time T.

Hint: Recall that the first fundamental theorem of calculus states that [itex]\frac{\mathrm{d} }{\mathrm{d} x} \int_{a}^{x}f(w)dw=f(x)[/itex].

2. Relevant equations

[itex]T=klog(1+au)[/itex], [itex]\frac{\mathrm{d} }{\mathrm{d} x}, \int_{a}^{x}f(w)dw=f(x)[/itex]

[itex]F(v)=m\frac{\mathrm{d}v }{\mathrm{d} t}[/itex]

3. The attempt at a solution

As far as I can see the best thing to do is take Newton's second law stated above, and separate the variables to give:

[itex]m\int \frac{dv}{F(v)}=\int_{0}^{T}dt[/itex]

Which would give simply T on the RHS, so substituting for T would give:

[itex]m\int \frac{dv}{F(v)}=klog(1+au)[/itex]

Now I am not sure how to proceed. I am tempted to try and take F(v) out of the integral to the other side of the equation, but I'm not sure whether this is allowed:

[itex]m\int \frac{dv}{klog(1+au)}=F(v)[/itex]

If so, then my answer would simply be [itex] \frac{mv}{klog(1+au)}=F(v)[/itex]. Any ideas as if this is correct?

For the second part I think I need to modify our starting point with Newton's second law to say:

[itex]mv\frac{\mathrm{d} v}{\mathrm{d} x}=F(v)[/itex]

But after that I am a bit lost. If I have the equation for F(v) perhaps I can separate variables again and then express v as dx/dt and integrate again. Unfortunately i cannot see far enough ahead to see how this would work.

Thanks everyone in advance for taking a look at this!

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# Homework Help: Particle motion in a straight line - find opposing force as a function of velocity

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