Using F=kvx in order to describe x as a function of t

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SUMMARY

The discussion focuses on deriving the position function x(t) for a particle subjected to a force defined by F=kvx, where k is a constant and v is the initial velocity at time t=0. The relationship between force, mass, and acceleration is established using Newton's second law, F=ma, leading to the differential equation dv/dx=kx/m. The solution process involves integrating both sides, resulting in the equation V=(kx^2/2m)+Vnaught, which is then used to find dx/dt and ultimately integrate to determine x as a function of time t.

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Homework Statement


The force acting on a particle of mass m is given by
F=kvx
in which k is a positive constant. The particle passes through the origin with the speed v naught at time t=0. Find x as a function of (t)


Homework Equations


F=ma
a=(dx/dt)(dv/dx)



The Attempt at a Solution


F=kvx

ma=kvx

a=kvx/m

(dx/dt)(dv/dx)=kvx/m

v(dv/dx)=kvx/m

dv/dx=kx/m

dv=(kx/m)dx

integral of both sides left in terms of dv and right in terms of dx

V-Vnaught=kx^2/2m

I'm not sure where to go from here. Help would be very much appreciated!
 
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Nevermind I figured it out. For anyone who's curious.

V=(kx^2/2m)+Vnaught
dx/dt=(kx^2/2m)+Vnaught
dx/((kx^2/2m)+Vnaught)=dt
then integrate both sides.
 
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