Solving Differential Problem: Constant Power, Mass Acceleration

[Cate]

Homework Statement

An automobile with mass m accelerates starting from rest, while the engine supplies constant power P.

a) Show that the speed is given as a function of time by v= (2Pt/m)^(1/2)

b) Show that the acceleration is given as a function of time by a= (P/2mt)^(1/2)

c) Show that the displacement is given as a function of time by x-x0= (8P/9m)^(1/2)t^(3/2)

The Attempt at a Solution

I know that this isn't a difficult question but I always seem to get confused. For a) I started trying to find the derivative v= (1/2)(2P/m) ^ (-1/2)...but I don;t know how to contiue by showing that it's a function of time. If someone could just help me with a I'm sure i'll be fine for b) and c)

Thanks!

 

[quantumdude]

You're trying to start the problem from the thing you're supposed to show. It isn't going to work.

One of the handful of expressions for power is P=Fv. You know that P is constant, which means that F and v must be functions of time. Since you want only one dependent variable (namely, v), here is your first subtask: Eliminate the variable F by expressing it in terms of mass and velocity. You should be able to do this using Newton's second law. Do you see what I mean?

 

[Cate]

I don't really understand what you mean, so Newton's second law ( F= Ma) In the place of P (for part a))Iinput FV then in the place of F I input Ma so it looks like V= (2ma(t)/m)^ (1/2) so the m's cancel out right? then what do I do?

 

[quantumdude]

I don't really understand what you mean, so Newton's second law ( F= Ma) In the place of P (for part a))

No, you're going to put it in place of F. And what is a, in terms of v?

so it looks like V= (2ma(t)/m)^ (1/2) so the m's cancel out right?

You won't get off that easy. You will end up with dv/dt in your equation, and you will have to integrate it to find v(t).