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

$a = k(120-v)$

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

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

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

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

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

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

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

$v(0) = 0$

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

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

$a = 120e^{-t}$

$96 = 120e^{-t}$

solve for t

3. ## Ans not right!!!

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

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