Calculating Force and Displacement for a Particle in Advanced Mechanics

[mch]

I'm new to this site, so please bear with me.

1. Homework Statement

A particle of mass m has speed v(x) = α/√x. Calculate the force F(x) responsible. Then, calculate the displacement x(t) of the particle.

Homework Equations

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The equations that I believe we are supposed to use are f=ma and f=m*dv/dt

The Attempt at a Solution

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Since F= m*dv/dt, i tried this:

F = m*dv/dx*dx/dt = mv*dv/dx = -mvα/(2(x)^(3/2))

And that's as far as I could get for the first problem. However, this seems off because the force is in terms of two variables, right? V and x?

 

[Orodruin]

You already know v as a function of x so it is really only one variable ...

 

[vela]

You're given an expression for v in terms of x...

 

[mch]

Oh okay great! So F(x) = -mα^2/(2x^2). Thank you! How silly of me.

So now I need to find x(t). Do i say that F = -mα^2/(2x^2) = mx'' and solve the second order differential equation? My initial conditions are that v(x=0) = 0 and x(t = 0)=0.

 

[Orodruin]

Those initial conditions are incompatible with v(x) = α/√x.

 

[Orodruin]

Also, you do not need to use the force equation, you already know that dx/dt = v(x) so you can just integrate this.

 

[mch]

Thank you. I figured this out and I believe I got all the right answers.

Thanks a lot for your help!