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
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 determine which statement is NOT true about the motion of a projectile, we will analyze each of the four statements provided:

1. **Time of flight of a projectile is proportional to the speed with which it is projected.**
- The formula for the time of flight (T) of a projectile is given by:
\[ T = \frac{2u \sin \theta}{g} \]
where \( u \) is the initial speed, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity.
- From this formula, we can see that the time of flight is indeed proportional to the initial speed \( u \).
- **Conclusion:** This statement is TRUE.

2. **The horizontal range of the projectile is proportional to the square root of the speed with which it is projected.**
- The formula for the horizontal range (R) of a projectile is given by:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
- From this formula, we can see that the range \( R \) is proportional to the square of the initial speed \( u \), not the square root.
- **Conclusion:** This statement is NOT TRUE.

3. **For a given speed of projection, the angle of projection for maximum range is 45 degrees.**
- To maximize the range, we use the sine function in the range formula. The maximum value of \( \sin 2\theta \) occurs when \( 2\theta = 90^\circ \) or \( \theta = 45^\circ \).
- **Conclusion:** This statement is TRUE.

4. **At maximum height, the acceleration due to gravity is perpendicular to the velocity of the projectile.**
- At maximum height, the vertical component of the velocity becomes zero, while the horizontal component remains. The acceleration due to gravity acts downward, while the horizontal component of velocity is directed horizontally.
- Since the vertical acceleration (gravity) is perpendicular to the horizontal velocity, this statement is also TRUE.
- **Conclusion:** This statement is TRUE.

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

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