Vertically launched rocket problem with integration

[Lawrencel2]

Homework Statement

Find the Rockets height as a function of time
It is in a constant field g. u refers to the exhaust speed and M is the initial mass.
Starts from rest. and is single stage.

Homework Equations

1) m * dv/dt= -dm/dt * u-mg

2)Show that height as t is: y(t)= u*t- 1/2*g*t^2- u*t*ln(M/m)

The Attempt at a Solution

Ok, i arrived at a function very similar to the the height but, i cannot seem to get the u*t term in the function. i get y(t)= - 1/2*g*t^2- u*t*ln(M/m)
where do i get the u*t term from? i feel like i am so close to the answer but so far away..
I integrated 1) and arrived at V(t)= u*ln(M/m)-g*t

 

[Lawrencel2]

no help?

 

[gash789]

Hi, I did look at this earlier but got stuck with your first integration. It is likely there is a constant of integration you are missing so if you are still stuck and want to, please can you show me your steps to integrate 1) and then I will see if I can help?

Cheers

 

[PICsmith]

I'm pretty sure you're forgetting that mass is a function of time, so your integration of u*ln(M/m(t))*dt isn't as simple as you had hoped...

Edit:
Hint: Do the integration w.r.t. mass, and not time. The key here is that (I'm assuming) the fuel burn rate is constant, so dm/dt = -c, where c is some constant. Use that to replace dt with -dm/c. Also, don't forget to solve for the constant of integration using the initial condition m(0) = M.

 

[paranormal]

Then i integrated again to get y(t)= u*t*ln(M/m)- 1/2*g*t^2 + C
But i cannot seem to get rid of the constant term to match the given function. Can someone please help me out?


It seems like you are on the right track with your approach. However, there may be a mistake in your integration process. When integrating the velocity equation, you should get V(t) = u*ln(M/m) - g*t + C. Then, when integrating again to find the height equation, you should get y(t) = u*t*ln(M/m) - 1/2*g*t^2 + C*t + D, where C and D are integration constants. To get rid of the constant terms, you can use the given initial conditions of the rocket starting from rest and the initial mass M to solve for the values of C and D. This should give you the final equation of y(t) = u*t*ln(M/m) - 1/2*g*t^2. I hope this helps!