- #1
RoyalCat
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Homework Statement
A steam-engine is traveling along a rail, with a constant power output of 1.5MW, regardless of its velocity.
1. What is the mass of the steam-engine if it is known that it accelerates from 10\frac{m}{s} to 25\frac{m}{s} in 6 sec?
2. Describe the velocity of the steam-engine as a function of time.
And the other questions follow from these two.
Homework Equations
P=\vec F \cdot \vec V=\frac{\Delta W}{\Delta t}=constant
The Attempt at a Solution
1.
P=\frac{\Delta W}{\Delta t}=\tfrac{1}{2}\frac{(v_f^2-v_0^2)}{\Delta t}m
m=\frac{2P\Delta t}{v_f^2-v_0^2}
//
2.
F=\frac{P}{v}
a=\tfrac{P}{m}v^{-1}
\dot v=\tfrac{P}{m}v^{-1} Here we have a simple differential equation.
\frac{dv}{dt}=\tfrac{P}{m}\frac{dt}{dx}
\frac{dv}{dx}\frac{dx}{dt}=\tfrac{P}{m}\frac{dt}{dx}
v\cdot dv=\tfrac{P}{m}dt
\tfrac{1}{2}v^2=\tfrac{P}{m}\cdot t+C
Now how do I get from here to the final solution? Omitting the +C completely provides a function v(t) that is indeed a solution to the differential equation. However, it assumes that v_0=0
I can't quite find a way to incorporate the initial values into the solution.
Well, I did find a way, but it doesn't quite make sense mathematically, I'd love a thorough explanation as to why the following is correct:
\tfrac{P}{m}\equiv \tau
v(t)=\sqrt{2\tau\cdot t+v_0^2}
Taking the time-derivative provides us with:
\dot v(t)=\frac{1}{2\sqrt{2\tau\cdot t+v_0^2}}\cdot 2\tau
The above must also be equal to: \tau v^{-1} in order to satisfy the differential equation.
\tau v^{-1}=\tau \frac{1}{\sqrt{2\tau\cdot t+v_0^2}}
So the differential equation is satisfied.
Checking that against the data in question #1, shows that it holds, and it makes sense, since at t=0\rightarrow v=v_0, however, it seems as though we used the integration constant quite haphazardly, so I'd love an explanation. :)
Is it just a matter of isolation v(t)=\sqrt{2\tau\cdot t+2C}
And then finding, using the initial values of the problem that 2C=v_0^2 ?With thanks in advance, Anatoli.
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