Rocket Sled Motion Equation and Free Response Models | Question Check

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mpm
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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.
 
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mpm said:
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.
 
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?
 
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.
 
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?