Animated Solution for Physics - Kinematics: A tennis ball is released from a height $h$ and after freely falling on a wooden floor, it rebounds and reaches height $h/2$. The velocity versus height of the ball during its motion may be represented graphically by (Graphs are drawn schematically and on not to scale)

Welcome, future engineers and physicists! Today, we are going to dive into a beautiful problem that perfectly marries physical intuition with graphical representation. We will analyze the motion of a bouncing tennis ball and translate its journey into a velocity-height graph.

Imagine you are holding a tennis ball at a height $h$ above a solid wooden floor. You release it from rest. The moment you let go, gravity takes over, and the ball begins its free fall.

To translate this physical motion into a mathematical graph, we need our trusty kinematic equations. Since we are interested in the relationship between velocity ($v$) and height ($y$), the third equation of motion is our best friend:

Let's focus on the fall. The ball starts from rest, so the initial velocity $u=0$. The acceleration is due to gravity, acting downwards, so $a=-g$. The displacement from the initial height $h$ to any height $y$ is $s=y-h$.

Substituting these into our equation, we get:

Notice that $v^2$ is directly proportional to the change in height. Mathematically, this means that if we plot $v$ on the y-axis and $h$ on the x-axis, the relationship $y=h-\frac{v^2}{2g}$ represents a parabola.

During the fall, the velocity is directed downwards. By standard convention, we take the downward direction as negative. So, as the height decreases from $h$ to $0$, the velocity becomes increasingly negative, starting from $0$ and reaching $-\sqrt{2gh}$ just before impact. On our graph, this traces a parabolic curve in the fourth quadrant, with an arrow pointing from $h$ towards the negative $v$-axis.

Now comes the exciting part—the collision! The ball hits the floor and rebounds. However, the problem states it only reaches a maximum height of $h/2$. This tells us that the collision was inelastic; the ball lost some kinetic energy to sound and heat.

For the upward journey, the ball starts at height $y=0$ with some positive initial velocity $v^{\prime }$, and comes to a momentary halt ($v=0$) at height $y=h/2$. Using the same kinematic equation:

As the ball rises, its height increases from $0$ to $h/2$, and its positive velocity decreases from $+\sqrt{gh}$ to $0$. This traces another parabolic curve, this time in the first quadrant. The arrow must point from the positive $v$-axis towards $h/2$ on the horizontal axis, indicating the forward flow of time.

When we compare our meticulously constructed graph with the given options, Option (c) is the only one that perfectly captures both the parabolic shapes and the correct directional arrows for the falling and rebounding phases.

Physics is not just about equations; it's about visualizing the story those equations tell. Keep visualizing, and keep conquering!