1. ## Differential equation

Hi,
I am kinda having trouble with the following problem:

Suppose that the acceleration of a model rocket is proportional to the difference between 120 ft/sec and the rocket's velocity. If it is initially at rest and its initial acceleration is 120 ft/sec^2, how long will it take to accelerate to 96 ft/s?
t = ______________ sec

What I have tried:

dv/dt=k(120-v)

integrating both sides:

-ln|120-v| = t + C

Since, t = 0 and initial velocity v = 0

Thus, C = -ln|120|

I am stuck here:

Many thanks....

2. Originally Posted by althaemenes
Hi,
I am kinda having trouble with the following problem:

Suppose that the acceleration of a model rocket is proportional to the difference between 120 ft/sec and the rocket's velocity. If it is initially at rest and its initial acceleration is 120 ft/sec^2, how long will it take to accelerate to 96 ft/s?
t = ______________ sec

$\displaystyle a = k(120-v)$

$\displaystyle a(0) = 120$ , $\displaystyle v(0) = 0$

$\displaystyle 120 = k(120)$ ... $\displaystyle k = 1$

$\displaystyle \frac{dv}{dt} = 120-v$

$\displaystyle \frac{-dv}{120-v} = -dt$

$\displaystyle \ln|120-v| = -t + C$

$\displaystyle 120-v = Ae^{-t}$

$\displaystyle v = 120 + Ae^{-t}$

$\displaystyle v(0) = 0$

$\displaystyle 0 = 120 + A$ ... $\displaystyle A = -120$

$\displaystyle v = 120(1 - e^{-t})$

$\displaystyle a = 120e^{-t}$

$\displaystyle 96 = 120e^{-t}$

solve for t

3. ## Ans not right!!!

Hi, for t I got 0.22314 or ln(120/96) but its not right....

$\displaystyle 96 = 120(1-e^{-t})$