A students have built a cart of mass M with an engine that can propel it with a non-constant force. In a test run, they launch the cart on a horizontal surface from rest. While the surface is horizontal, the cart’s engine provides a force F(x)=Fo+cx3 in the direction of the cart’s motion, where x is the distance traveled, and Fo and c are positive constants. Due to a design flaw, the engine stops working after the cart has traveled a distance D, as soon as the cart encounters a frictionless incline that makes an angle θ with the horizontal.

Use the Work-Kinetic Energy Theorem to derive an expression for the speed of the cart when it has reached a vertical height H above the starting point.