Question

a ball is thrown vertically upwards and comes down again. air resistance is negligible. which of the following graphs shows how the gravitational potential energy $e_p$ varies with time $t$?

Explanation:

Step1: Recall potential - energy formula

The gravitational potential energy of an object of mass $m$ at height $h$ above the ground is given by $E_p=mgh$. When the ball is thrown vertically upwards, its height $h$ first increases, reaches a maximum, and then decreases as it falls back down.

Step2: Analyze the motion of the ball

The height - time relationship of a ball thrown vertically upwards with initial velocity $u$ is a quadratic function $h = ut-\frac{1}{2}gt^{2}$ (using the kinematic equation $s = ut+\frac{1}{2}at^{2}$, where $s = h$, $a=-g$). Since $E_p=mgh$, $E_p$ is also a quadratic function of time. At $t = 0$, $h = 0$ so $E_p=0$. As $t$ increases, $h$ increases and so does $E_p$. At the maximum - height, the velocity of the ball is zero, and then as the ball falls back down, $h$ decreases and so does $E_p$.

Step3: Match with the graph

The graph of a quadratic function is a parabola. The gravitational potential energy starts from zero, increases to a maximum value at the maximum height of the ball, and then decreases back to zero when the ball returns to the initial position.

Answer:

D.