A 0.5 kg ball is thrown up into the air with an initial speed of 9 m/s. At what height does the gravitational potential energy of the ball equal its initial kinetic energy?

A. When the ball goes up and comes back down to the same height that it was initially thrown from.
B. The gravitational potential energy of the ball never equals its initial kinetic energy.
C. When the ball reaches its maximum height, just before it starts to come back down.
D. When the ball is halfway up to its maximum height.

The gravitational potential energy of a 0.5 kg ball, thrown up with an initial speed of 9 m/s, equals its initial kinetic energy when the ball reaches its maximum height. At this point, the ball has no upward speed, and its potential energy, dependent on the height, equals the initial kinetic energy.

Explanation:

The question asks when does the gravitational potential energy of a 0.5 kg ball, thrown up in the air with an initial speed of 9 m/s, equal its initial kinetic energy. The answer is c) When the ball reaches its maximum height, just before it starts to come back down. It's because, at this point, the kinetic energy (linked to the speed of the ball) is minimised (ideally, zero for an instant), and all the initial kinetic energy has been converted into potential energy due to the height.

To illustrate this, let's first calculate the initial kinetic energy using the formula: Kinetic Energy = 0.5 * mass * velocity^2. Substituting the given values, we get 0.5 * 0.5 kg * (9 m/s)^2 = 20.25 J (Joules). The gravitational potential energy at a certain height is given by the formula: Potential Energy = mass * g * height, with g being the gravitational acceleration (approx. 9.8 m/s^2). To find out when this energy equals the initial kinetic energy, we set 20.25 J = 0.5 kg * 9.8 m/s^2 * h and solve for h. The sought height comes out to be approximately 4.12 m.

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