Answer
$t=0.518~s$
Work Step by Step
The required time can be determined as follows:
$\Sigma F_y=0$
$\implies P-2Tsin\theta=0$
$\implies P-(2\times 5000\times \frac{4}{5})=0$
$\implies P=8000lb$
$m=\frac{W}{g}=\frac{5000}{32.2}=155.27Slugs$
Now we apply the impulse momentum principle in the vertical direction
$mv_1+\Sigma \int_{t_1}^{t_2} F_y dt=mv_2$
We plug in the known values to obtain:
$155.27(0)+\int_0^t 8000dt-\int_0^t 5000 dt=155.27(10)$
This simplifies to:
$t=0.518~s$
