A body is projected at an angle of `30^(@)` with the horizontal with momentum `P`.At its highest point the magnitude of the momentum is:

Options

A. `sqrt3/2P`
B. `2/sqrt3P`
C. `P`
D. `P/2`

Explanation

To solve the problem of finding the magnitude of the momentum of a body projected at an angle of \(30^\circ\) with the horizontal at its highest point, we can follow these steps:

### Step 1: Understand the Initial Momentum
The body is projected with an initial momentum \(P\). The momentum \(P\) can be expressed in terms of mass \(m\) and velocity \(v\) as:
\[ P = m \cdot v \]

### Step 2: Break Down the Initial Velocity
When the body is projected at an angle of \(30^\circ\), we can resolve the initial velocity \(v\) into horizontal and vertical components:
- Horizontal component: \(v_x = v \cos(30^\circ)\)
- Vertical component: \(v_y = v \sin(30^\circ)\)

### Step 3: Analyze the Highest Point
At the highest point of the projectile's trajectory, the vertical component of the velocity becomes zero (\(v_y = 0\)), while the horizontal component remains unchanged:
\[ v_x = v \cos(30^\circ) \]

### Step 4: Calculate the Horizontal Momentum at the Highest Point
The momentum at the highest point is only due to the horizontal component because the vertical momentum is zero. Therefore, the momentum \(P_{highest}\) at the highest point can be expressed as:
\[ P_{highest} = m \cdot v_x = m \cdot (v \cos(30^\circ)) \]

### Step 5: Relate the Initial Momentum to the Horizontal Momentum
Since \(P = m \cdot v\), we can substitute \(m\) from the initial momentum:
\[ m = \frac{P}{v} \]
Substituting this into the equation for \(P_{highest}\):
\[ P_{highest} = \frac{P}{v} \cdot (v \cos(30^\circ)) = P \cos(30^\circ) \]

### Step 6: Calculate \(\cos(30^\circ)\)
The value of \(\cos(30^\circ)\) is \(\frac{\sqrt{3}}{2}\). Therefore, we can substitute this value into our equation:
\[ P_{highest} = P \cdot \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the magnitude of the momentum at the highest point is:
\[ P_{highest} = \frac{P \sqrt{3}}{2} \]