A ball thrown vertically upward with initial velocity u. Which of the following graphs represent the velocity of ball correctly as function of time?
(Take upward direction as positive)

Options

A.
B.
C.
D.

Answer

To determine the correct graph representing the velocity of a ball thrown vertically upward with an initial velocity \( u \), we can analyze the motion step by step.

### Step 1: Understand the motion of the ball
When a ball is thrown upward, it moves against the force of gravity. The initial velocity \( u \) is positive (upward direction), and as the ball rises, its velocity decreases due to gravitational acceleration \( g \) acting downwards.

**Hint:** Remember that gravity acts in the opposite direction to the initial velocity.

### Step 2: Write the equation for velocity
The velocity \( v \) of the ball at any time \( t \) can be expressed using the equation:
\[ v = u - gt \]
where:
- \( v \) is the velocity at time \( t \),
- \( u \) is the initial velocity,
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( t \) is the time in seconds.

**Hint:** The negative sign in the equation indicates that gravity is acting in the opposite direction to the initial velocity.

### Step 3: Analyze the velocity over time
1. At \( t = 0 \), the velocity \( v = u \) (the ball is at its maximum upward speed).
2. As time increases, the term \( gt \) increases, causing the velocity \( v \) to decrease.
3. The ball will reach its maximum height when \( v = 0 \). Setting the equation to zero:
\[ 0 = u - gt \implies t = \frac{u}{g} \]
4. After reaching the maximum height, the ball will start to fall back down, and the velocity will become negative (downward direction).

**Hint:** The point where the velocity becomes zero corresponds to the maximum height of the ball.

### Step 4: Sketch the graph
- The graph starts at \( v = u \) (positive value).
- The velocity decreases linearly to zero at \( t = \frac{u}{g} \).
- After reaching zero, the velocity becomes negative and continues to decrease (in magnitude) as the ball falls back down.

**Hint:** The graph should show a straight line starting from a positive value, decreasing to zero, and then continuing downward into the negative region.

### Conclusion
The correct graph representing the velocity of the ball as a function of time will start at \( u \), decrease linearly to zero, and then continue into the negative region as the ball falls back down.

**Final Answer:** The correct option is **C**, which represents this behavior accurately.