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 c?
would the bicyclists potential energy be higher at b or c?
would the bicyclists total energy be higher at b or c?
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: Analyze potential energy

Potential energy \(U = mgh\) (where \(m\) is mass, \(g\) is acceleration due to gravity, \(h\) is height). The higher the position, the higher the potential energy. Point \(A\) is the highest, so highest potential energy at \(A\).

Step2: Analyze lowest potential energy

Lowest height means lowest potential energy. Point \(C\) is the lowest, so lowest potential energy at \(C\).

Step3: Analyze kinetic energy

Kinetic energy \(K=\frac{1}{2}mv^{2}\). When potential energy is lowest (by conservation of mechanical energy \(E = K + U\), assuming no non - conservative forces), kinetic energy is highest. At \(C\), \(U\) is lowest, so \(K\) is highest at \(C\).

Step4: Analyze highest speed

Since \(K=\frac{1}{2}mv^{2}\), highest \(K\) means highest \(v\). So highest speed at \(C\).

Step5: Compare \(K\) at \(B\) and \(C\)

As \(U\) at \(B>U\) at \(C\) (because height at \(B >\) height at \(C\)), by \(E=K + U\) (constant), \(K\) at \(C>K\) at \(B\).

Step6: Compare \(U\) at \(B\) and \(C\)

Since \(h\) at \(B>h\) at \(C\) and \(U = mgh\), \(U\) at \(B>U\) at \(C\).

Step7: Compare total energy at \(B\) and \(C\)

In the absence of non - conservative forces (no friction, no air resistance), total mechanical energy \(E = K+U\) is conserved. So \(E\) at \(B = E\) at \(C\).

Step8: Determine farthest point without pedaling

By conservation of mechanical energy, the cyclist can reach the same height as the starting point. Point \(A\) and \(E\) are at the same height. But since she starts from \(A\) and moves down, the farthest point without pedaling is \(E\).

Answer:

• Where would the bicyclist have the highest potential energy? \(A\)
• Where would the bicyclist have the lowest potential energy? \(C\)
• Where would the bicyclist have the highest kinetic energy? \(C\)
• Where would the bicyclist have the highest speed? \(C\)
• Would the bicyclist's kinetic energy be higher at \(B\) or \(C\)? \(C\)
• Would the bicyclist's potential energy be higher at \(B\) or \(C\)? \(B\)
• Would the bicyclist's total energy be higher at \(B\) or \(C\)? Neither (they are equal)
• 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? \(E\)