Find the angle of projection of a projectile for which the horizontal range and maximum height are equal.

Explanation

To find the angle of projection of a projectile for which the horizontal range and maximum height are equal, we can follow these steps:

### Step 1: Write the formulas for range and maximum height
The formulas for the horizontal range \( R \) and maximum height \( H \) of a projectile are given by:
- **Range**:
\[ R = \frac{u^2 \sin 2\theta}{g} \]
- **Maximum Height**:
\[ H = \frac{u^2 \sin^2 \theta}{2g} \]

### Step 2: Set the range equal to the maximum height
According to the problem, we need to set the horizontal range equal to the maximum height:
\[ R = H \]
Substituting the formulas, we have:
\[ \frac{u^2 \sin 2\theta}{g} = \frac{u^2 \sin^2 \theta}{2g} \]

### Step 3: Simplify the equation
We can cancel \( u^2 \) and \( g \) from both sides (assuming \( u \neq 0 \) and \( g \neq 0 \)):
\[ \sin 2\theta = \frac{1}{2} \sin^2 \theta \]

### Step 4: Use the double angle identity
Recall that \( \sin 2\theta = 2 \sin \theta \cos \theta \). Substituting this into the equation gives:
\[ 2 \sin \theta \cos \theta = \frac{1}{2} \sin^2 \theta \]

### Step 5: Rearrange the equation
Rearranging the equation leads to:
\[ 4 \sin \theta \cos \theta = \sin^2 \theta \]
or
\[ \sin^2 \theta - 4 \sin \theta \cos \theta = 0 \]

### Step 6: Factor the equation
Factoring out \( \sin \theta \):
\[ \sin \theta (\sin \theta - 4 \cos \theta) = 0 \]

### Step 7: Solve for \( \theta \)
This gives us two cases:
1. \( \sin \theta = 0 \) (which corresponds to \( \theta = 0^\circ \) or \( \theta = 180^\circ \), not valid for projectile motion)
2. \( \sin \theta - 4 \cos \theta = 0 \)
\[ \sin \theta = 4 \cos \theta \]
Dividing both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ \tan \theta = 4 \]

### Step 8: Find the angle
Now, we can find the angle \( \theta \):
\[ \theta = \tan^{-1}(4) \]

### Final Answer
Thus, the angle of projection for which the horizontal range and maximum height are equal is:
\[ \theta = \tan^{-1}(4) \]