Which one of the following statements is NOT true about the motion of a projectile ?

Options

A. The time of flight of a projectile is proportional to the speed with which it is projected at a given angle of projection
B. The horizontal range of a projectile is proportional to the square root of the speed with which it is . Projected
C. For a given speed of projection, the angle of projection for maximum range is `45^(@)`
D. At maximum height, the acceleration due to gravity is perpendicular to the velocity of the projectile

Answer

To solve the question, we need to analyze each statement about the motion of a projectile and determine which one is NOT true.

### Step-by-Step Solution:

1. **Understanding Projectile Motion**:
- A projectile is an object that is thrown into the air with an initial velocity and follows a curved path under the influence of gravity. The motion can be analyzed in two dimensions: horizontal and vertical.

2. **Evaluating the First Statement**:
- **Statement**: "The time of flight of a projectile is proportional to the speed with which it is projected at a given angle of projection."
- **Analysis**: The time of flight \( T \) is given by the formula:
\[ T = \frac{2u \sin \theta}{g} \]
where \( u \) is the initial velocity, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of projection. This shows that time of flight is indeed proportional to the initial speed \( u \).
- **Conclusion**: This statement is TRUE.

3. **Evaluating the Second Statement**:
- **Statement**: "The horizontal range of a projectile is proportional to the square root of the speed with which it is projected."
- **Analysis**: The range \( R \) is given by the formula:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
Here, the range is proportional to the square of the initial speed \( u \), not the square root.
- **Conclusion**: This statement is FALSE.

4. **Evaluating the Third Statement**:
- **Statement**: "For a given speed of projection, the angle of projection for maximum range is 45 degrees."
- **Analysis**: To achieve maximum range, the angle of projection should indeed be 45 degrees. This can be derived from the range formula where \( \sin(2\theta) \) is maximized at \( \theta = 45^\circ \).
- **Conclusion**: This statement is TRUE.

5. **Evaluating the Fourth Statement**:
- **Statement**: "At maximum height, the acceleration due to gravity is perpendicular to the velocity of the projectile."
- **Analysis**: At maximum height, the vertical component of the velocity becomes zero, while the acceleration due to gravity acts downwards. The velocity at this point is horizontal, making the acceleration due to gravity perpendicular to the velocity.
- **Conclusion**: This statement is TRUE.

### Final Conclusion:
The statement that is NOT true about the motion of a projectile is the **second statement**: "The horizontal range of a projectile is proportional to the square root of the speed with which it is projected."