Question

Just c please. The answer for b is 39.6 m/s

P11 A rock is dropped from a cliff of height 80 m into the ocean. Disregard air resistance. Once it hits the water, the rock's acceleration is a -1.5 s1 v, where v is the rock's velocity. a) b) c) How long does it take to hit the water? What is its velocity when it hits the water? Find、(1), the time dependence of the rock's velocity inside the water.

Answer 1

(c) Once it hits water the acceleration is given by,

$a(t) = -1.5 v$

Therefore force is given by,

$F = ma \Rightarrow \frac{\mathrm{d} v}{\mathrm{d} t} = -1.5v$

Rearranging we get,

$\Rightarrow -\frac{\mathrm{d} v}{1.5v} = \mathrm{d}t$

We know the initial condition is 39.6 m/s at t = 0, Therefore integrating with limits we get,

$\Rightarrow -\int_{39.6}^{v}\frac{\mathrm{d} v}{1.5v} = \int_0^t \mathrm{d}t$

$\Rightarrow - \frac{1}{1.5}\int_{39.6}^{v}\frac{\mathrm{d} v}{v} = \int_0^t \mathrm{d}t$

$\Rightarrow - \frac{1}{1.5}\left | ln (v) \right |_{39.6}^v = t$

$\Rightarrow - \frac{1}{1.5} ln \left(\frac{v}{39.6} \right ) = t$

$\Rightarrow v(t) = 39.6e^{-1.5t}$