Which of the following is the altitude-time graph for a projectile thrown horizontally from the top of the tower

Options

A.
B.
C.
D.

Answer

To determine the altitude-time graph for a projectile thrown horizontally from the top of a tower, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- We have a tower of height \( H \).
- A projectile is thrown horizontally from the top of the tower with an initial horizontal velocity \( u_0 \).
- The initial vertical velocity of the projectile is \( 0 \) (since it is thrown horizontally).

2. **Vertical Motion Analysis**:
- The only force acting on the projectile in the vertical direction is gravity, which causes a downward acceleration \( g \).
- The vertical displacement \( s \) of the projectile after time \( t \) can be described by the equation:
\[ s = \frac{1}{2} g t^2 \]
- Since the projectile starts from height \( H \), the height \( h(t) \) at any time \( t \) can be expressed as:
\[ h(t) = H - \frac{1}{2} g t^2 \]

3. **Graph Characteristics**:
- At \( t = 0 \), the height \( h(0) = H \).
- As time increases, the height decreases due to the downward motion caused by gravity.
- The projectile will eventually reach the ground when \( h(t) = 0 \). Setting the height equation to zero gives:
\[ 0 = H - \frac{1}{2} g t^2 \implies t^2 = \frac{2H}{g} \implies t = \sqrt{\frac{2H}{g}} \]
- The graph of \( h(t) \) versus \( t \) will be a downward-opening parabola, starting from \( H \) and reaching \( 0 \) at \( t = \sqrt{\frac{2H}{g}} \).

4. **Identifying the Correct Graph**:
- The correct altitude-time graph should start at height \( H \) and curve downwards, eventually reaching \( 0 \) at a specific time.
- Among the options provided, we need to identify the one that matches this description.

5. **Conclusion**:
- The correct altitude-time graph for a projectile thrown horizontally from the top of a tower is the one that starts at height \( H \) and follows a parabolic path down to \( 0 \).

### Final Answer:
The correct option is the one that represents a downward-opening parabola starting from height \( H \) and reaching \( 0 \) at a certain time.

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