(eight-pound object dropped from a height of 5000 feet, where air resistance is proportional to the velocity.)
write the velocity as a function of time if its velocity after 5 secs is approximately -101 feet per second.
(eight-pound object dropped from a height of 5000 feet, where air resistance is proportional to the velocity.)
write the velocity as a function of time if its velocity after 5 secs is approximately -101 feet per second.
True, but don't blame us - we use SI units, and have done for 40 years or so! (All right, they are a bit French, but we all have to swallow our pride at some time or other!)
In fact, slugs are horrible creatures, best avoided. Just call the mass $\displaystyle m$ - it cancels anyway.
The equation of motion is then:
$\displaystyle -mg + k'v = m\frac{dv}{dt}$, where $\displaystyle v$ is measured vertically upwards
And you can then divide through by $\displaystyle m$, and replace $\displaystyle \frac{k'}{m}$ by $\displaystyle k$ to obtain:
$\displaystyle -g + kv = \frac{dv}{dt}$
After separating the variables, integrating, and using $\displaystyle v(0) = 0$, this gives:
$\displaystyle v = \frac{g}{k}(1 - e^{kt})$
So when $\displaystyle t = 5, v=-101$ (and $\displaystyle g = 32$):
$\displaystyle -101 = \frac{32}{k}(1 - e^{5k})$
How you are expected to solve this equation for $\displaystyle k$, I don't know (Newton's method?). In fact, I found $\displaystyle k \approx -0.2$ (but I cheated and used a spreadsheet!)
So the solution is
$\displaystyle v\approx 160(e^{-0.2t}-1)$
Grandad