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Some modern roller coasters are "launched" rather than slowly pulled to the top of a hill. One recent design is a coaster on a U-shaped track like that shown above. Each time the coaster passes through the acceleration zone, it receives a kick from an electric motor. Suppose that for the track shown above, the height H is 75 m. The acceleration zone has a width d = 25 m. If a coaster of mass m = 11,000 kg experiences an accelerating horizontal force of magnitude 0.5mg while it is in the acceleration zone, how many passes through the zone must the coaster make before it will reach the top of the track? (After this, the function of the motor will be reversed to decelerate the coaster.)

Solution

Model: Mechanical Energy and Non-Conservative Work.

Approach: We begin with an initial-state final-state diagram and energy bar graphs.

It is clear from the diagrammatical representations that the system has gained mechanical energy. The source of the energy is the electric motor. In equation form, we have:

\begin{large}\[ E_{i} + W^{NC} = 0 + W^{NC} = E_{f} = mgH\]\end{large}

where the non-conservative work is due to the motor.

From the definition of work, the work done by the motor can be related to the force provided by the motor:

\begin{large}\[ W^{NC}_{\rm motor} = \sum_{passes} F_{\rm motor} d = nF_{\rm motor}d\]\end{large}

where n is the number of passes the train makes through the acceleration zone. Since the force provided by the motor is 0.5mg, we can write:

\begin{large}\[ \frac{1}{2}nmgd = mgH \]\end{large}

which is solved to obtain:

\begin{large}\[  n = \frac{2H}{d} = 6 \]\end{large}

which implies the coaster must make 6 complete passes through the acceleraation zone to reach the top of the track.