A body is projected at an angle `60^(@)` with the horizontal ground with kinetic energy k. When the velocity makes an angle `30^(@)` with the horizontal, the kinetic energy of the body will be `(k)/(a)`. Find value of a.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the initial conditions
A body is projected at an angle of \(60^\circ\) with the horizontal with an initial kinetic energy \(K\). The initial velocity can be expressed as:
\[ K = \frac{1}{2} m u^2 \]
where \(u\) is the initial speed of the body and \(m\) is its mass.

### Step 2: Analyze the velocity components
When the body reaches a point where the velocity makes an angle of \(30^\circ\) with the horizontal, we need to find the new velocity \(v\) at that point. The horizontal component of the velocity remains constant throughout the projectile motion.

The horizontal component of the initial velocity is:
\[ u_x = u \cos(60^\circ) = u \cdot \frac{1}{2} = \frac{u}{2} \]

At the point where the angle is \(30^\circ\), the horizontal component of the velocity is:
\[ v_x = v \cos(30^\circ) = v \cdot \frac{\sqrt{3}}{2} \]

### Step 3: Set the horizontal components equal
Since the horizontal component of velocity remains constant, we can set the two expressions for horizontal velocity equal to each other:
\[ \frac{u}{2} = v \cdot \frac{\sqrt{3}}{2} \]

### Step 4: Solve for \(v\)
Rearranging the equation gives:
\[ v = \frac{u}{\sqrt{3}} \]

### Step 5: Calculate the new kinetic energy
The kinetic energy \(K'\) at the point where the velocity makes an angle of \(30^\circ\) is given by:
\[ K' = \frac{1}{2} m v^2 \]
Substituting \(v = \frac{u}{\sqrt{3}}\):
\[ K' = \frac{1}{2} m \left(\frac{u}{\sqrt{3}}\right)^2 = \frac{1}{2} m \cdot \frac{u^2}{3} = \frac{1}{3} \left(\frac{1}{2} m u^2\right) = \frac{K}{3} \]

### Step 6: Relate \(K'\) to \(K\)
We know that \(K' = \frac{K}{a}\). From our previous result, we have:
\[ \frac{K}{3} = \frac{K}{a} \]

### Step 7: Solve for \(a\)
By comparing both sides, we find:
\[ a = 3 \]

### Final Answer
The value of \(a\) is \(3\).

---