Prove that the maximum horizontal range is four times the maximum height attained by the projectile, when fired at an inclination so as to have maximum horizontal range.

Explanation

To prove that the maximum horizontal range (R) is four times the maximum height (H) attained by a projectile when fired at an inclination for maximum range, we can follow these steps:

### Step 1: Define the equations for range and height
The horizontal range (R) of a projectile launched with an initial velocity \( u \) at an angle \( \theta \) is given by the formula:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
The maximum height (H) attained by the projectile is given by:
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]

### Step 2: Determine the angle for maximum range
To find the angle that gives the maximum range, we need to maximize \( \sin 2\theta \). The maximum value of \( \sin 2\theta \) is 1, which occurs when \( 2\theta = 90^\circ \) or \( \theta = 45^\circ \).

### Step 3: Calculate the maximum range at \( \theta = 45^\circ \)
Substituting \( \theta = 45^\circ \) into the range formula:
\[ R = \frac{u^2 \sin 90^\circ}{g} = \frac{u^2}{g} \]

### Step 4: Calculate the maximum height at \( \theta = 45^\circ \)
Now, substituting \( \theta = 45^\circ \) into the height formula:
\[ H = \frac{u^2 \sin^2 45^\circ}{2g} = \frac{u^2 \left(\frac{1}{\sqrt{2}}\right)^2}{2g} = \frac{u^2 \cdot \frac{1}{2}}{2g} = \frac{u^2}{4g} \]

### Step 5: Relate maximum range to maximum height
Now, we can relate the maximum range \( R \) to the maximum height \( H \):
\[ R = \frac{u^2}{g} \]
\[ H = \frac{u^2}{4g} \]

### Step 6: Show that \( R = 4H \)
To show that the maximum horizontal range is four times the maximum height, we can express \( R \) in terms of \( H \):
\[ R = 4H \]
Substituting the expression for \( H \):
\[ R = 4 \left(\frac{u^2}{4g}\right) = \frac{u^2}{g} \]

### Conclusion
Thus, we have proved that the maximum horizontal range \( R \) is four times the maximum height \( H \) attained by the projectile when fired at an inclination of \( 45^\circ \).