Answer
$v=20.2ft/s$
$h=6.36ft$
Work Step by Step
We can determine the required speed and the distance as follows:
We know that:
$\Sigma F_y=0$
$\implies N_cos\theta=0$
$\implies N-(800cos 9.04)=0$
$\implies N=790.06lb$
First, we calculate the tangential velocity of the car
$mr_1v_1+\int_0 ^t rNsin\theta=mrv_t$
$\implies 0+\int_0^4 (8\times 790.06sin 9.04)dt=(\frac{800}{32.2})8v_t$
$\implies v_t=19.99ft/s$
Now we determine the final speed of the car
$v_2=\frac{v_t}{cos\theta}$
$\implies v_2=\frac{19.99}{cos 9.04}$
$\implies v_2=20.2ft/s$
According to the principle of work and energy
$\frac{1}{2}mv_1^2+Wh=\frac{1}{2}mv_2^2$
We plug in the known values to obtain:
$\frac{1}{2}(\frac{800}{32.2})(0)^2+800h=\frac{1}{2}(\frac{800}{32.2})(20.2)^2$
This simplifies to:
$h=6.36ft$
