A particle is projected with a velocity u making an angle `theta` with the horizontal. At any instant its velocity becomes v which is perpendicular to the initial velocity u. Then v is

Options

A. `u cos theta`
B. `u tan theta`
C. `u cot theta`
D. `u sec theta`

Explanation

To solve the problem, we need to analyze the motion of a particle projected at an angle θ with an initial velocity u. The key point is that at some instant, the velocity v of the particle becomes perpendicular to the initial velocity u. We will derive the expression for v step by step.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- A particle is projected with an initial velocity \( u \) at an angle \( \theta \) with the horizontal.
- The initial velocity can be resolved into two components:
- Horizontal component: \( u_x = u \cos \theta \)
- Vertical component: \( u_y = u \sin \theta \)

2. **Condition for Perpendicular Velocities**:
- At a certain instant, the velocity \( v \) of the particle becomes perpendicular to the initial velocity \( u \).
- If the angle between the initial velocity \( u \) and the velocity \( v \) is 90 degrees, then the angle between the horizontal and the new velocity \( v \) can be expressed as \( 90 - \theta \).

3. **Resolving the New Velocity**:
- The new velocity \( v \) can also be resolved into components:
- Horizontal component: \( v_x = v \cos(90 - \theta) = v \sin \theta \)
- Vertical component: \( v_y = v \sin(90 - \theta) = v \cos \theta \)

4. **Using the Horizontal Component**:
- Since there is no acceleration in the horizontal direction (assuming no air resistance), the horizontal component of the velocity remains constant:
\[ u_x = v_x \]
Therefore,
\[ u \cos \theta = v \sin \theta \]

5. **Solving for v**:
- Rearranging the equation gives:
\[ v = \frac{u \cos \theta}{\sin \theta} \]
- This can be simplified using the cotangent function:
\[ v = u \cot \theta \]

### Final Result:
Thus, the expression for \( v \) when it is perpendicular to the initial velocity \( u \) is:
\[ v = u \cot \theta \]