Rocket Sled Motion Equation and Free Response Models | Question Check

[mpm]

Ive got 2 questions that I would like looked at.

Question 1:
A rocket sled has the following equation of motion: 6vdot = 2700 - 24*v. How long must the rock fire before the sled travels 2000 m? The sled starts from rest.

I took the integral which of that equation which gamve me v(t) = 2700*t - 24*x. At rest v = 0. So 0 = 2700t - 24*(2000)

Solve for t and you get t = 17.78 seconds.

Does this look right? If not please let me know.

Question 2:
For each of the following models, obtain the free response and time constants if any.

16*xdot + 14*x = 0, x(0) = 6

I changed it to v's, which gave me 16*v + 14*vdot = 0, v(0) = 6

For time constant its c/m so tau = 14/16 = .875

Then for the free response its v(t) = v(0)*e^-t/tau

So for my final answer, v(t) = 6*e^-1.143*t

If there are any problems with this, can you please let me know where.

I just want to make sure I am doing this right.

 

[LeonhardEuler]

Ive got 2 questions that I would like looked at.
Question 1:
A rocket sled has the following equation of motion: 6vdot = 2700 - 24*v. How long must the rock fire before the sled travels 2000 m? The sled starts from rest.
I took the integral which of that equation which gamve me v(t) = 2700*t - 24*x. At rest v = 0. So 0 = 2700t - 24*(2000)
Solve for t and you get t = 17.78 seconds.
Does this look right? If not please let me know.

The question says that the sled starts from rest, not that it is at rest after it has traveled 2000m.

 

[mpm]

So I should have a variable still I guess for my v(t) instead of setting it to equal 0. Can you give me an idea of what it might equal?

 

[LeonhardEuler]

I would go with another approach entirely: Solve the differential equation for v, then integrate to get an expression for x(t) with no v terms.

 

[mpm]

I did it a different way for Problem 1.

I made the equation into 6*vdot + 24*v = 2700.

Then I solved for V(t) which is the free response and force response.

Since v(0) = 0, the free response goes away.

leaving v(t) = F/c(1-e^(-ct/m) = 2700/4*(1-e^-4t)

I then took the integral of that to get x(t).

This game me x(t) = (225*e^-4t*(4t*e^(4t)+1))/8

I then set x(t) = 2000. Solving for t I come up with 17.78 seconds which is what I originaly came up with the first time.

Was I just right the first time? Is this coincidental or did I do it wrong this time?