(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

An object of mass M begins with a velocity of 0 m/s at a point. A power input of P watts goes directly to kinetic energy until the object has traveled a distance of X meters. What is the final velocity of the object?

So, we have constant variables

M = mass

X = distance that power will be input

P = power level

And also

V = final velocity, after traveling distance X (to be solved)

2. Relevant equations

E = 1/2MV[tex]^{2}[/tex]

X = [tex]\int[/tex]V(t) dt - V(t) is V as a function of time.

E = P*t

3. The attempt at a solution

Find V as a function of E (easy)

V = [tex]\sqrt{2E}[/tex]/M

Find V as a funciton of time t (use equation)

V = [tex]\sqrt{2P*t}[/tex]/M

Now take the equation for X

X = [tex]\int[/tex]V(t) dt

And Find X as a function of t directly, knowing the V(t) function

X = [tex]\int[/tex][tex]\sqrt{2P*t}[/tex]/M dtEdit: Should be, and was calculated as([tex]\int[/tex] [tex]\sqrt{2P*t}[/tex] dt)/M

Integrate (remember that sqrt(2P) is a constant)

X(t) = (2/3)*[tex]\sqrt{2P}[/tex]*t[tex]^{3/2}[/tex]/M

Now change to t in terms of X

t(X) = ((3/2)/[tex]\sqrt{2P}[/tex]*M*X)[tex]^{2/3}[/tex]

And finally slide that into the V(t) equation

V(X) = [tex]\sqrt{2P}[/tex]*((3/2)/[tex]\sqrt{2P}[/tex]*M*X)[tex]^{1/3}[/tex]/M

Simplify (whew!)

V(X) = 3[tex]^{1/3}[/tex]*2[tex]^{-1/6}[/tex]*P[tex]^{1/3}[/tex]*X[tex]^{1/3}[/tex]*M[tex]^{-2/3}[/tex]

So, V correlates directly with the cube root of X, the cube root of P, and M^(-2/3), with a weird constant.

Am I right? Am I not? If not, where did I go wrong?

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# Homework Help: Final velocity of an object accelerated across a given distance with constant power

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