A ball is projected vertically up with an initial velocity. Which of the following graphs represents the KE of the ball?

Options

A.
B.
C.
D.

Answer

To solve the problem of determining which graph represents the kinetic energy (KE) of a ball projected vertically upward, we can follow these steps:

### Step 1: Understand the motion of the ball
When a ball is projected vertically upward with an initial velocity \( U_0 \), it will rise to a certain height until its velocity becomes zero at the peak of its motion. After reaching the peak, it will start to fall back down.

### Step 2: Analyze kinetic energy
The kinetic energy (KE) of the ball at any height \( h \) can be expressed using the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( v \) is the velocity of the ball at height \( h \).

### Step 3: Relate velocity to height
Using the kinematic equation:
\[ v^2 = U_0^2 - 2gh \]
where \( g \) is the acceleration due to gravity (acting downward), we can express the kinetic energy in terms of height \( h \):
\[ KE = \frac{1}{2} m (U_0^2 - 2gh) \]
This simplifies to:
\[ KE = \frac{1}{2} m U_0^2 - mgh \]

### Step 4: Identify the behavior of KE
1. At the initial point (when \( h = 0 \)), the KE is at its maximum value of \( \frac{1}{2} m U_0^2 \).
2. As the ball rises, \( h \) increases, which means \( KE \) decreases because of the \( -mgh \) term.
3. At the peak of its motion (maximum height), the velocity \( v = 0 \) and thus \( KE = 0 \).
4. As the ball falls back down, \( h \) decreases, causing \( KE \) to increase again until it reaches the initial value just before hitting the ground.

### Step 5: Graphical representation
From the analysis:
- The graph of KE will start at a maximum value when \( h = 0 \), decrease to zero at the peak height, and then increase again as the ball falls back down.
- This behavior indicates that the graph of KE will be a downward-opening parabola, starting from a positive value, reaching zero, and then increasing again.

### Step 6: Identify the correct graph
Based on the characteristics of the kinetic energy as described:
- The graph must start high (maximum KE) at \( h = 0 \), decrease to zero at the peak height, and then increase again as the ball descends.

Thus, the correct option that represents the kinetic energy of the ball is **Option C**.