Question $2$

$[$Total Marks: $8]$

Consider a torpedo experiencing drag that slows it down as the torpedo moves through the ocean. Let the torpedo velocity be $v(t).$ The rate of change of velocity is then

$\frac{dv}{(d)t}$

Suppose that the rate of change of velocity is proportional to the square of velocity, $v^2.$

$($a$)$ Write the relationship between $d\frac{v}{d}t$ and $v^2$ using the symbol prop. Hence, write down a differential equation for the torpedo velocity. Use $k,$ where $k>0,$ to denote any constants of proportionality.

$[2]$

$($b$)$ If the initial speed of the torpedo is $v_0,$ write down the initial condition.

$[1]$

$($c$)$ Solve your differential equation, using your initial condition to find the constant of integration.

$[4]$

$($d$)$ What happens to the torpedo velocity as $t?$

$[1]$