When you throw an object into the air, is its return speed just before hitting your hand the same as its initial speed? (Neglect air resistance.) Explain by applying the conservation of mechanical energy.

a. Yes. The kinetic energy converts to an equal amount of potential energy at the peak of the throw, which converts back to the same amount of kinetic energy at the bottom of the throw. The initial and final kinetic energies are equal, so the initial and return speeds are equal.

b. Yes. Potential energy is gained before reaching the peak of the throw, and lost after the peak of the throw, but the kinetic energy never changes. Because the kinetic energy is always the same, the initial and return speeds are equal.

c. No. During the throw, the object gains potential energy depending on the height of the peak of the throw. Because it has more potential energy when it lands than when it is thrown, it has less kinetic energy when it lands, and so the initial and return speeds are not equal.

d. No. During the throw, the object gains potential energy depending on the height of the peak of the throw. This potential energy is added to the initial kinetic energy at the end of the throw, so the object has more kinetic energy when it lands, and so the initial and return speeds are not equal.