A body is projected at `30^(@)` with the horizontal. The air offers resistance in proportional to the veclocity of the body. Which of the following statements is correct?

Options

A. The trajectroy is a symmetrical parabola
B. the time of rise to the maximum height is equal to the time of return to the ground
C. The velocity at the highest pint is directed along the horizontal
D. the sum of the kinetic and potential energies remains constant

Explanation

To solve the problem, let's analyze the motion of the body projected at an angle of \(30^\circ\) with the horizontal, considering the effect of air resistance that is proportional to the velocity of the body.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The body is projected at an angle of \(30^\circ\) with an initial velocity \(u\).
- The initial velocity can be resolved into horizontal and vertical components:
- Horizontal component: \(u_x = u \cos(30^\circ) = u \cdot \frac{\sqrt{3}}{2}\)
- Vertical component: \(u_y = u \sin(30^\circ) = u \cdot \frac{1}{2}\)

2. **Effect of Air Resistance**:
- The air resistance acts in the opposite direction to the velocity and is proportional to the velocity. This means:
- In the horizontal direction, the air resistance is \(R_x = -k u_x\) (where \(k\) is the proportionality constant).
- In the vertical direction, the air resistance is \(R_y = -k u_y\).

3. **Equations of Motion**:
- The equations of motion for the horizontal and vertical components can be written as:
- Horizontal:
\[ \frac{du_x}{dt} = -k u_x \]
- Vertical:
\[ \frac{du_y}{dt} = -g - k u_y \]
where \(g\) is the acceleration due to gravity.

4. **Behavior of the Velocity**:
- As the body moves upward, the vertical component of the velocity \(u_y\) decreases due to the combined effect of gravity and air resistance. Eventually, \(u_y\) will reach zero at the highest point of the trajectory.
- The horizontal component \(u_x\) will also decrease due to air resistance, but it will not become zero until the body stops moving.

5. **At the Highest Point**:
- At the highest point of the trajectory, the vertical component of the velocity \(u_y\) is zero. However, the horizontal component \(u_x\) will still be present but reduced due to air resistance.

6. **Conclusion**:
- The correct statement regarding the motion of the body is that at the highest point, the vertical velocity is zero, and only the horizontal velocity remains, which is affected by air resistance.

### Final Answer:
At the highest point of the projectile's trajectory, the vertical velocity is zero, and the horizontal velocity is still present but reduced due to air resistance.