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## Homework Statement

a body with mass M moves across a plane with friction

friction constant:

[itex]\mu = \lambda x^2[/itex]

the body starts at x=0

with velocity v

_{0}

find at what x

the body stops

and what was the velocity half way there.

## Homework Equations

[itex]v^2=v_0^2+2a\Delta x[/itex]

## The Attempt at a Solution

obviously,

[itex]F(x)=mg\mu = mg\lambda x^2[/itex]

so

[itex]a(x)=g \lambda x^2[/itex]

so in the equation [itex]v^2=v_0^2+2a\Delta x[/itex]

I get

[itex]v^2=v_0^2+2g\lambda x^3[/itex]

the Question is, can I use this equation? the acceleration is not constant and this equation

depend on the fact that [itex]x=v_0t+ \frac{a}{2}t^2[/itex]

and [itex]v=v_0+at[/itex]

(and it's not true for non-constant acceleration)

if I cant, how can I integrate the acceleration?

or how do I get v(x)?

Thanks.

EDIT:

I used [itex] a= v\frac{dv}{dx}[/itex]

therefore

[itex]vdv=adx[/itex]

[itex]\int_{v_0}^{v(x)}{vdv} = g\lambda \int_{0}^{x}{x^2}[/itex]

[itex]\frac{1}{2} ( v(x)^2- v_0^2) =\frac{1}{3} g\lambda x^3[/itex]

[itex]v(x)^2=v_0^2+\frac{2}{3}g\lambda x^3[/itex]

does that seem right?

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