A body is projected horizontally with a velocity of `u ms^(-1)` at an angle `theta` with the horizontal. The kinetic energy at intial the highest point is `(3)/(4)th` of the kinetic energy. The value of `theta` is :

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `120^(@)`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have a body projected at an angle \(\theta\) with an initial velocity \(u\). At the highest point of its trajectory, the kinetic energy is given to be \(\frac{3}{4}\) of the initial kinetic energy.

### Step 2: Write the Expression for Initial Kinetic Energy
The initial kinetic energy (\(KE_o\)) when the body is projected is given by the formula:
\[ KE_o = \frac{1}{2} m u^2 \]
where \(m\) is the mass of the body and \(u\) is the initial velocity.

### Step 3: Determine the Velocity at the Highest Point
At the highest point of the projectile's motion, the vertical component of the velocity becomes zero, and only the horizontal component remains. The horizontal component of the velocity can be expressed as:
\[ v_{x} = u \cos \theta \]
Thus, the kinetic energy at the highest point (\(KE_a\)) is:
\[ KE_a = \frac{1}{2} m (u \cos \theta)^2 = \frac{1}{2} m u^2 \cos^2 \theta \]

### Step 4: Set Up the Equation Relating the Kinetic Energies
According to the problem, the kinetic energy at the highest point is \(\frac{3}{4}\) of the initial kinetic energy:
\[ KE_a = \frac{3}{4} KE_o \]
Substituting the expressions for kinetic energy, we have:
\[ \frac{1}{2} m u^2 \cos^2 \theta = \frac{3}{4} \left(\frac{1}{2} m u^2\right) \]

### Step 5: Simplify the Equation
We can cancel \(\frac{1}{2} m u^2\) from both sides of the equation (assuming \(m \neq 0\) and \(u \neq 0\)):
\[ \cos^2 \theta = \frac{3}{4} \]

### Step 6: Solve for \(\cos \theta\)
Taking the square root of both sides gives:
\[ \cos \theta = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Step 7: Find the Angle \(\theta\)
The angle \(\theta\) for which \(\cos \theta = \frac{\sqrt{3}}{2}\) is:
\[ \theta = 30^\circ \]

### Final Answer
Thus, the value of \(\theta\) is:
\[ \theta = 30^\circ \]

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