Question

a bicyclist is stopped at the entrance to a valley, as sketched below:
where would the bicyclist have the highest potential energy?
where would the bicyclist have the lowest potential energy?
where would the bicyclist have the highest kinetic energy?
where would the bicyclist have the highest speed?
would the bicyclists kinetic energy be higher at b or a?
would the bicyclists potential energy be higher at b or a?
would the bicyclists total energy be higher at b or a?
suppose the bicyclist lets off the brakes and coasts down into the valley without pedaling. even if there is no friction or air resistance to slow her down, what is the farthest point the bicyclist could reach without pedaling?

Explanation:

Step1: Potential Energy and Height

Potential energy \(U = mgh\) (where \(m\) is mass, \(g\) is acceleration due to gravity, \(h\) is height). Higher the height, higher the potential energy. Point \(A\) and \(F\) are at the highest positions among the marked points. Assuming the cyclist starts from rest at \(A\) (initial point), and the sketch likely has \(A\) as the starting - highest point (if we consider the cyclist's journey from \(A\) down the valley). The lowest point is \(D\), so \(U_{min}\) at \(D\) (\(h\) is minimum at \(D\)).

Step2: Kinetic Energy and Speed

Kinetic energy \(K=\frac{1}{2}mv^{2}\). At the lowest point (\(D\)), potential energy is converted into kinetic energy (by conservation of mechanical energy \(E = K + U\), assuming no non - conservative forces like friction). So \(K_{max}\) at \(D\), and since \(K=\frac{1}{2}mv^{2}\), \(v_{max}\) at \(D\) (when \(K\) is maximum).

Step3: Comparing \(A\) and \(B\)

At \(A\), \(K = 0\) (cyclist is stopped). As the cyclist moves from \(A\) to \(B\), \(h\) decreases (\(U=mgh\) decreases) and \(K=\frac{1}{2}mv^{2}\) increases. So \(K_{B}>K_{A}\), \(U_{A}>U_{B}\).

Step4: Total Energy

By conservation of mechanical energy \(E = K + U\) (in the absence of non - conservative forces like friction and air resistance), \(E_{A}=E_{B}\).

Step5: Farthest Point

By conservation of mechanical energy (no non - conservative forces), the cyclist can reach a point at the same height as the starting point. So the farthest point is \(F\).

Answer:

• Highest potential energy: \(A\)
• Lowest potential energy: \(D\)
• Highest kinetic energy: \(D\)
• Highest speed: \(D\)
• Kinetic energy (\(B\) vs \(A\)): \(B\)
• Potential energy (\(B\) vs \(A\)): \(A\)
• Total energy (\(B\) vs \(A\)): Equal
• Farthest point: \(F\)