ODE question

Mcarbitrage

Any assistance would be more than welcome, thanks!


A particle P of mass m moves with speed v such that
its motion is governed by the ODE:

m(dv/dt) = −mg − γv^2

where g, γ > 0 are constants. The initial condition for P is that
v(t = 0) = U > 0. In the subsequent motion, v decreases as
the particle moves, until v = 0 after the particle has moved a
distance H from its starting position.
Find the distance H

Blackpanther95

Cross multiply . Let u^2 be the denominator of the fraction. So we have 1/u^2dv/dt. now dv/dt =du/dt times some constants blah blah. anyways the key thing is we get d(-1/u)/dt essiantially and this can be solved easily

Blackpanther95

I forgot to mention you should use the chain rule: dv/dt=v*dv/ds. Then let u be the RHS find dv/ds in terms of u and du/ds... Nice cancellations and then an answer with logs. Hope that helps

Blackpanther95

May I ask where you got this equation out of; it is very difficult!

Mcarbitrage

Its a first year Maths degree 'homework' question

Blackpanther95

Mcarbitrage wrote: »

Its a first year Maths degree 'homework' question

The lecturer likes seeing people suffer, I see. Do you understand my solution?

Mcarbitrage

I'm sorry I cant quite follow your solution. Have u solved it completely? If so do you mind writing up your solution? Thanks!

locum-motion

From the Charter:

"5)This forum has a high number of users who are students, either at secondary or at third level. It is considered to be against the spirit of this forum to provide said users with full solutions to homework/exam type questions. The onus should be on posters to provide a hint or encouragement to the user so that they can attempt a solution on their own"



Thread reported.

Mcarbitrage

Sorry, wasn't aware of this, didn't want full solution with answer, more the order in which he went about solving this as couldn't completely understand what he was saying previously.

Blackpanther95

Okay, so; 1)Think about a suitable substitution for dv/dt, using the chain rule express this as v*dv/ds (details ommited)
2)Let u = to the RHS of the original equation
3) differentiate u with respect to s and on the other side you should have -2yvdv/ds
4)Go back to the original equation and replace the dv/dt with the result from 3, and replace the RHS with u (that was the substitution)
5) The resulting differential equation is linear and first order, the solution is simple (it should be if you've done it right)
6) Solve the equation, and back-substitute for u.
7) s=s(t) is the displacement; for the question asked there is no need to find v=v(t) ; v=v(s) will suffice.

Mcarbitrage

Thanks, I finally got it!