Final answer:
The kinetic energy of a bowling ball rolling up a ramp is converted into potential energy. Using energy conservation, we find the ball's velocity at the top is 2.0 m/s for 0.5 m high ramp and it does make it. For a 1 m high ramp, it does not have enough energy to reach the top.
Step-by-step explanation:
The question involves converting the initial kinetic energy of the bowling ball into gravitational potential energy as it rolls up the ramp without slipping. According to the conservation of energy, the initial kinetic energy plus the initial potential energy equals the final kinetic energy plus the final potential energy. In this case, as the bowling ball rolls up the ramp, its kinetic energy decreases while its potential energy increases.
(a) For a ramp 0.5 m high, we can use the following energy conservation equation:
KE_initial + PE_initial = KE_top + PE_top. The initial potential energy is zero since we start from ground level, thus we have the following equation: 0.5 x m x v^2 = 0.5 x m x v_top^2 + m x g x h. Given that the mass (m) cancels out and the acceleration due to gravity (g) is 9.8 m/s^2, we solve for the velocity at the top (v_top).
(b) For a ramp that is 1 m high, we also use the energy conservation equation to determine whether the ball makes it to the top. If the final kinetic energy at the top of the ramp is negative, it means the ball does not have enough energy to reach the top and hence it supposedly will not make it.
By calculating the energies for both scenarios, we can determine the correct answer: a) v = 2.0 m/s, Yes for the first case and c) v = 3.2 m/s, No for the second case indicating that for the higher ramp the bowling ball doesn't make it to the top.