A particle is projected from ground at some angle with the horizontal. Let `P` be the point at maximum height `H`. At what height above the point `P` should the particle be aimed to have range equal to maximum height ?

Options

A. H
B. 2 H
C. H//2
D. 3 H

Explanation

To solve the problem, we need to determine the height above point P (the maximum height H) to which the particle should be aimed so that the range equals the maximum height. Let's break it down step by step.

### Step 1: Understand the Problem
We know that a particle is projected at an angle and reaches a maximum height \( H \). We need to find a height \( h \) above point P such that when the particle is aimed at this height, the range \( R \) of the projectile equals \( H \).

### Step 2: Relate Range and Maximum Height
From the projectile motion equations, we know that the range \( R \) of a projectile is given by:
\[ R = \frac{v^2 \sin(2\theta)}{g} \]
And the maximum height \( H \) is given by:
\[ H = \frac{v^2 \sin^2(\theta)}{2g} \]
Given that we want \( R = H \), we can set these two equations equal to each other.

### Step 3: Use the Relationship Between Range and Height
Since we know that \( R = H \), we can express this as:
\[ \frac{v^2 \sin(2\theta)}{g} = \frac{v^2 \sin^2(\theta)}{2g} \]
This simplifies to:
\[ \sin(2\theta) = \frac{1}{2} \sin^2(\theta) \]

### Step 4: Find the Angle
Using the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \), we can rewrite the equation:
\[ 2 \sin(\theta) \cos(\theta) = \frac{1}{2} \sin^2(\theta) \]
This can be rearranged to find \( \tan(\theta) \).

### Step 5: Relate Height Above Point P
Let \( C \) be the point where the particle should be aimed. We can express the height \( h \) above point P as:
\[ h = H + PC \]
Where \( PC \) is the vertical distance from point P to point C.

### Step 6: Set Up the Equation
From the geometry of the projectile motion, we can relate the distances:
\[ R = 2H \]
This means that the height \( h \) above point P should equal the maximum height \( H \).

### Step 7: Conclusion
Thus, the height \( h \) above point P is equal to the maximum height \( H \):
\[ h = H \]

### Final Answer
The height above point P should be equal to the maximum height \( H \).

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