- #1

Alabaster

- 3

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## Homework Statement

The kinetic energy K of an object of mass m and velocity v is given by K=1/2mv

^{2}. Suppose an object of mass m, moving in a straight line, is acted upon by force F=F(s) that depends on position s. According to Newton's Second Law F(s)=ma. A is acceleration of the object.

Show that the work done from moving a position s

_{0}to a position s

_{1}is equal to the change in the object's kinetic energy; that is, show that

Work=[itex]\int^{s1}_{s0}[/itex]F(s)ds=[itex]\frac{1}{2}[/itex]mv[itex]^{2}_{1}[/itex]-[itex]\frac{1}{2}[/itex]mv[itex]^{2}_{0}[/itex]

where v

_{0}=v(s

_{0}) and v

_{1}=v(s

_{1}) are the velocities of the objects at positions s

_{0}and s

_{1}.

## Homework Equations

I think all of the relevant equations are listed above.

## The Attempt at a Solution

Work=[itex]\int^{s1}_{s0}[/itex]F(s)ds=[itex]\frac{1}{2}[/itex]mv[itex]^{2}_{1}[/itex]-[itex]\frac{1}{2}[/itex]mv[itex]^{2}_{0}[/itex]

Looking at the right side of the equation suggests that I take the integral and it splits into two, since it's a definite integral. So I should start with this Work=[itex]\int^{s1}_{s0}[/itex]F(s)ds and mess with it until it's the same as the right side of the equation.

This is where I get stuck. I asked a guy for help and he showed me

ma=m[itex]\frac{dv}{dt}[/itex]

and then said "s" is a displacement vector s=vdt

He showed me the next step, which was =[itex]\int^{t1}_{t0}[/itex]Fvdt

I'm very confused how he got there, but he showed me the whole problem. This is it in its entirety.

Work=[itex]\int^{s1}_{s0}[/itex]F(s)ds

=[itex]\int^{t1}_{t0}[/itex]Fvdt

=[itex]\int^{t1}_{t0}[/itex]m[itex]\frac{dv}{dt}vdt[/itex]

=m[itex]\int^{t1}_{t0}[/itex]vdv

=m(v

^{2}/2) evaluated from t

_{0}to t

_{1}

=[itex]\frac{1}{2}[/itex]mv[itex]^{2}_{1}[/itex]-[itex]\frac{1}{2}[/itex]mv[itex]^{2}_{0}[/itex]

If someone could explain what happened step-by-step, that would be amazing. I can follow some of the steps, but all the variable swapping is making this impossible for me to understand.

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