HalfThere
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Homework Statement
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)
Homework Equations
E = 1/2MV^{2}
X = \intV(t) dt - V(t) is V as a function of time.
E = P*t
The Attempt at a Solution
Find V as a function of E (easy)
V = \sqrt{2E}/M
Find V as a funciton of time t (use equation)
V = \sqrt{2P*t}/M
Now take the equation for X
X = \intV(t) dt
And Find X as a function of t directly, knowing the V(t) function
X = \int\sqrt{2P*t}/M dt Edit: Should be, and was calculated as (\int \sqrt{2P*t} dt)/M
Integrate (remember that sqrt(2P) is a constant)
X(t) = (2/3)*\sqrt{2P}*t^{3/2}/M
Now change to t in terms of X
t(X) = ((3/2)/\sqrt{2P}*M*X)^{2/3}
And finally slide that into the V(t) equation
V(X) = \sqrt{2P}*((3/2)/\sqrt{2P}*M*X)^{1/3}/M
Simplify (whew!)
V(X) = 3^{1/3}*2^{-1/6}*P^{1/3}*X^{1/3}*M^{-2/3}
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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