- #1

derravaragh

- 24

- 0

## Homework Statement

A particle of mass m slides down an inclined plane under the influence of gravity. If the motion is resisted by a force f = kmv^2, show that the time required to move a distance d after starting from rest is

t = [arccosh(e^(kd))]/√(kgsin(θ)

where θ is the angle of inclination of the plane.

## Homework Equations

F = ma

F_g = mgsin(θ)

Resistance = -kmv^2

Motion of particle => ma = mgsin(θ) - kmv^2

## The Attempt at a Solution

My attempt was to set up the equation for the motion (ma = mgsin(θ) - kmv^2) and use differential equations to solve. After dividing by mass, I had:

dv/dt = gsin(θ) - kv^2

which I then divided by k, and substituted (g/k)sinθ for C^2 giving

dv/kdt = C^2 - v^2

After collecting terms and integrating I came to

t = arctan(v/√((g/k)sinθ))/√(kgsinθ)

I thought I was on the right track as I have the numerator correct, but I do not know where to go from here, or if this is even correct so far. Any help would be much appreciated. Also, I know this question has been asked before, but the answer given didn't make sense to me so it doesn't help.