Question

a ball is released at the left end. assume it has only enough friction to roll, but not to lessen its speed
rank from greatest to least, at each point the balls:
a. momentum
b. kinetic energy
c. potential energy

Explanation:

Momentum

• Momentum \(p = mv\). Since mass \(m\) is constant, momentum depends on velocity \(v\).
• At point \(C\), the ball is at the lowest point. By conservation of mechanical energy \(E = mgh+\frac{1}{2}mv^{2}\) (where \(h\) is height), at \(C\) potential energy \(mgh\) is minimum, so kinetic energy \(\frac{1}{2}mv^{2}\) is maximum, and thus velocity \(v\) is maximum.
• At point \(A\), the ball is released from rest (initial velocity \(v = 0\) at \(A\) in the context of the track - although it starts to move down, at the very point \(A\) (the left - most point as per the release), its velocity is the least among the non - moving or moving points considered in terms of the track's points. At \(D\), the height is higher than \(C\) but lower than \(A\), so \(v_D>v_A\) but \(v_D < v_C\). At \(B\), height is higher than \(C\) but lower than \(D\) (assuming the track's height order from the figure's general shape), so \(v_C>v_B > v_D>v_A\).
• So the order of momentum \(p_C>p_B > p_D>p_A\)

Kinetic Energy

• Kinetic energy \(K=\frac{1}{2}mv^{2}\). Since \(m\) is constant, \(K\propto v^{2}\)
• Using the same velocity analysis as for momentum (because \(K\) and \(p\) both depend on \(v\) with \(K\) depending on \(v^{2}\) and \(p\) on \(v\), and the mass is constant). The order of kinetic energy is \(K_C>K_B > K_D>K_A\)

Potential Energy

• Potential energy \(U = mgh\). Since \(m\) is constant and \(g\) is constant (\(g = 9.8\ m/s^{2}\)), \(U\propto h\)
• At point \(A\), the height \(h\) is the greatest. At point \(C\), the height \(h\) is the least. At point \(D\), the height is higher than \(C\) but lower than \(A\). At point \(B\), the height is higher than \(C\) but lower than \(D\) (assuming the track's height order from the figure's general shape). So the order of potential energy \(U_A>U_D > U_B>U_C\)

Answer:

A. \(C>B > D>A\)
B. \(C>B > D>A\)
C. \(A>D > B>C\)