- #1
RoyalCat
- 671
- 2
Homework Statement
A steam-engine is traveling along a rail, with a constant power output of [tex]1.5MW[/tex], regardless of its velocity.
1. What is the mass of the steam-engine if it is known that it accelerates from [tex]10\frac{m}{s}[/tex] to [tex]25\frac{m}{s}[/tex] in [tex]6 sec[/tex]?
2. Describe the velocity of the steam-engine as a function of time.
And the other questions follow from these two.
Homework Equations
[tex]P=\vec F \cdot \vec V=\frac{\Delta W}{\Delta t}=constant[/tex]
The Attempt at a Solution
1.
[tex]P=\frac{\Delta W}{\Delta t}=\tfrac{1}{2}\frac{(v_f^2-v_0^2)}{\Delta t}m[/tex]
[tex]m=\frac{2P\Delta t}{v_f^2-v_0^2}[/tex]
//
2.
[tex]F=\frac{P}{v}[/tex]
[tex]a=\tfrac{P}{m}v^{-1}[/tex]
[tex]\dot v=\tfrac{P}{m}v^{-1}[/tex] Here we have a simple differential equation.
[tex]\frac{dv}{dt}=\tfrac{P}{m}\frac{dt}{dx}[/tex]
[tex]\frac{dv}{dx}\frac{dx}{dt}=\tfrac{P}{m}\frac{dt}{dx}[/tex]
[tex]v\cdot dv=\tfrac{P}{m}dt[/tex]
[tex]\tfrac{1}{2}v^2=\tfrac{P}{m}\cdot t+C[/tex]
Now how do I get from here to the final solution? Omitting the [tex]+C[/tex] completely provides a function [tex]v(t)[/tex] that is indeed a solution to the differential equation. However, it assumes that [tex]v_0=0[/tex]
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:
[tex]\tfrac{P}{m}\equiv \tau[/tex]
[tex]v(t)=\sqrt{2\tau\cdot t+v_0^2}[/tex]
Taking the time-derivative provides us with:
[tex]\dot v(t)=\frac{1}{2\sqrt{2\tau\cdot t+v_0^2}}\cdot 2\tau[/tex]
The above must also be equal to: [tex]\tau v^{-1}[/tex] in order to satisfy the differential equation.
[tex]\tau v^{-1}=\tau \frac{1}{\sqrt{2\tau\cdot t+v_0^2}}[/tex]
So the differential equation is satisfied.
Checking that against the data in question #1, shows that it holds, and it makes sense, since at [tex]t=0\rightarrow v=v_0[/tex], 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 [tex]v(t)=\sqrt{2\tau\cdot t+2C}[/tex]
And then finding, using the initial values of the problem that [tex]2C=v_0^2[/tex] ?With thanks in advance, Anatoli.
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