Question

your science class goes on a field trip to the nearby amusement park for an assignment. before riding the roller coaster, your teacher has you observe other riders.
part a
at what point will the roller coaster car have transformed the greatest amount of gravitational potential energy into kinetic energy?

Explanation:

Step1: Recall the relationship between gravitational potential energy and kinetic energy

Gravitational potential energy \(U = mgh\) (where \(m\) is mass, \(g\) is acceleration due to gravity, \(h\) is height) and kinetic energy \(K=\frac{1}{2}mv^{2}\). According to the conservation of mechanical energy (neglecting non - conservative forces like friction in a simple analysis), \(E = U + K\) (total mechanical energy is constant). As the roller - coaster car descends, \(h\) (height) decreases, so \(U\) decreases and \(K\) increases. The greater the decrease in \(h\) (the steeper the descent from a higher point), the more \(U\) is converted into \(K\).

Step2: Analyze the height of each point

Point \(W\) is the highest point among \(W\), \(X\), \(Y\), and \(Z\). When the car moves from \(W\) to \(X\), \(Y\), or \(Z\), the change in height (\(\Delta h\)) from \(W\) is larger compared to starting from other points (e.g., if we consider the conversion from \(W\) to \(Z\), the vertical drop from \(W\) to \(Z\) is greater than from \(X\) to \(Z\) or \(Y\) to \(Z\)).

Answer:

The roller - coaster car will have the greatest amount of gravitational potential energy converted into kinetic energy at point \(Z\) (assuming the car moves from the highest point \(W\) to \(Z\) as the maximum vertical drop among the possible paths shown in the diagram. If we consider the conversion process, the larger the decrease in height (from the highest initial point \(W\) to the lowest point \(Z\) among the given options), the more gravitational potential energy is converted into kinetic energy).