Question

1. look at the roller coaster diagram below (points a-e).

the roller coaster starts at point a and moves along the
track. using what you know about potential and kinetic
energy, describe how the energy of the roller coaster
changes as it travels from a → b → c → d → e.
in your response, explain:
where the potential energy is greatest and why.
where the kinetic energy is greatest and why.
how energy is transformed between potential and kinetic
forms along the track.
you may reference specific points (a-e) in your
explanation.

Explanation:

• Potential Energy: Potential energy (\(PE = mgh\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(h\) is height) depends on height. At point \(A\), the roller - coaster is at the highest position among \(A - E\). So, \(PE\) is greatest at \(A\) because \(h\) is maximum at \(A\) (assuming \(m\) and \(g\) are constant).
• Kinetic Energy: Kinetic energy (\(KE=\frac{1}{2}mv^{2}\)) depends on speed. At point \(D\), the roller - coaster has likely gained the most speed (converting potential energy to kinetic energy as it descends from \(A\)). So, \(KE\) is greatest at \(D\) as speed \(v\) is maximum there (assuming \(m\) is constant).
• Energy Transformation:
• \(A - B\): As the roller - coaster moves from \(A\) to \(B\), it descends. So, potential energy decreases (\(\Delta PE=mg\Delta h\), \(\Delta h<0\)) and kinetic energy increases (\(\Delta KE=\frac{1}{2}m\Delta(v^{2})\), \(\Delta(v^{2})>0\)) since \(PE\) is converted to \(KE\) (\(\Delta PE+\Delta KE+\Delta E_{\text{other}} = 0\), assuming \(\Delta E_{\text{other}}\) (e.g., due to friction) is small over a short distance \(A - B\)).
• \(B - C\): It ascends from \(B\) to \(C\). So, \(PE\) increases (\(\Delta h>0\)) and \(KE\) decreases (\(\Delta(v^{2})<0\)) as \(KE\) is converted to \(PE\).
• \(C - D\): It descends from \(C\) to \(D\). \(PE\) decreases (\(\Delta h<0\)) and \(KE\) increases (\(\Delta(v^{2})>0\)) as \(PE\) is converted to \(KE\).
• \(D - E\): It ascends from \(D\) to \(E\). \(PE\) increases (\(\Delta h>0\)) and \(KE\) decreases (\(\Delta(v^{2})<0\)) as \(KE\) is converted to \(PE\).

Answer:

• Potential energy is greatest at \(A\) because it has the maximum height. Kinetic energy is greatest at \(D\) because it has the maximum speed.
• \(A - B\): \(PE\) decreases, \(KE\) increases (\(PE\to KE\)). \(B - C\): \(KE\) decreases, \(PE\) increases (\(KE\to PE\)). \(C - D\): \(PE\) decreases, \(KE\) increases (\(PE\to KE\)). \(D - E\): \(KE\) decreases, \(PE\) increases (\(KE\to PE\)).