Energy Problem (The Roller Coaster)

A roller coaster travels on a frictionless track as shown in Fig 5.31. (a) If the speed of the roller coaster at point A is 5.0m/s, what is its speed at point B? (b) Will it reach point C? (c) What minimum speed at point A is required for the roller coaster to reach point C?

To create the bar graphs, the knowledge of what K, Ug, and Us is in order.  While Ug represents the height of the object, K symbolizes the potential energy and Us represents a spring energy.  During point A, the initial energy graph needs an equal amount of K and Ug.  This is because the roller coaster is 5.0meters off the ground and therefore has the potential energy, Ug, to go down the hill.  There is no spring throughout this ride.  At point B, the intermediate energy the height, Ug, is gone and transferred to potential energy, K.  Since the roller coaster started out with 5.0m/s velocity and at the top of a hill, the coaster now needs all the energy it lost from its height, Ug, in potential energy to make it up the second hill.  By point C, the final energy graph represents the transfer of all the energy to Ug, height.  The coaster is now at 8.0 meters in height, which is taller than the first hill.  The potential energy it had at points A and B were used to propel the roller coaster to point C.

The speed at point B can be calculated with the equations K, potential energy, equals 1/2 mass multiplied by velocity squared and Ug, height, equals mass multiplied by gravity, 9.8m/s, and height, in meters.  Since the mass of the coaster isn't given, the combination of the equations need to be used to find the speed at point B.  If 1/2 mass times velocity squared plus mass times gravity times height is put equal to 1/2 mass times velocity at point B squared, the velocity can be determined.  Because the mass isn't given, one can begin by dividing the entire equation by mass.  Then, plug in the numbers-1/2 (5m/s) squared + (9.8m/s) (5m) equals 1/2 B's velocity squared.  B's velocity is then found to be 11.09m/s.

To find if the coaster reaches point C, the right side of the equation above is used equaled to the height equation, Ug equals gravity times height times mass.  Again, divide the entire equation by mass.  Plug in the numbers, 1/2 (5m/s) squared + (9.8m/s) (5m) equals (9.8m/s) times height.  By solving for height, the height comes out to be 6.28m.  The coaster doesn't reach point C at 8meters.

Finally to locate the minimum speed, velocity, required to reach point C from point A, use the equation from the last paragraph keeping the velocity as the variable, 1/2(v) squared + (9.8) (5m) ='s (9.8) (8).  The height of A is used in the first half of the equation because the coaster starts here and ends at 8meters.  That's why 8meters is used in the second half of the equation. The velocity needed at point A to reach 8meters at point C is 7.67m/s.