**
***People began to make roller coasters around the early 1900s. These early roller coasters were made of wood, and people learned how to construct them using the principles of physics and by a certain amount of experimentation.*

*Beginning in the 1980s, constructers of roller coasters began to use computers to design them. In this way they were able to create a great many designs and simulate them, thus finding a roller coaster's weakest points and determining the cost of making them fail-safe. This, and the practice of making them of steel, made the new roller coasters larger, safer, and more fun.*

*Nevertheless, a fair amount of knowledge about a roller coaster can be learned by applying simple physics without the aid of a computer. For the most part, the two forces acting on the car are gravity and the normal force, for we will ignore the force of friction during the ride. A force due to a motor carries the car to the top of the first hill. Finally there is a frictional force due to rubber bumpers pressing against the car which serve to stop it at the end of the ride so that other riders can get on.*

*The feeling a rider experiences in the car is related to the force exerted by the car's seat on his body perpendicular (normal) to the car's motion. This is expressed as a number of g's, where g is the acceleration due to gravity. For instance, if the rider (mass m) experiences a force 2mg, then he is said to experience 2g's.*

*Consider the figure below, a very simple roller coaster, in answering the following questions. A motor brings the car from point A to B, where it is has very little velocity at a height H _{1} above the ground. The slope of the first hill is an angle*

*θ*

*from*

*the horizontal. The loop is a circle of whose highest point is H*

_{2}above the ground. The mass of the car is M. The velocity of the car at point F is v_{F}.** **

What is the velocity of the car at point F, v_{F}?