#### Rocket Car

At time $$t=0$$, the rocket car of mass $$m$$ shown below is at location $$s=0$$, with zero speed. At this instant, the parking break is released, and the car begins rolling downhill. The hill is at an angle $$\theta$$ relative to horizontal as shown. At time $$t = t_a$$, the driver turns on the rocket which immediately produces a constant thrust $$F_T$$.

Assume that the thrust produced by the rocket is sufficiently large that the car eventually travels back uphill and passes the original starting point. Given $$m$$, $$\theta$$, $$F_T$$, $$t_a$$, and gravitational field strength $$g$$, find the following. (1) The time, $$t_d$$, it takes for the car to switch directions and start heading back uphill; (2) the distance, $$d$$ the car travels downhill before turning around.

You can find a qualitative/graphical solution to this problem here. An analytical problem-solving solution is provided below.