A particle is projected with velocity u at angle `theta` with horizontal. Find the time when velocity vector is perpendicular to initial velocity vector.

Explanation

To solve the problem of finding the time when the velocity vector of a projectile is perpendicular to its initial velocity vector, we can follow these steps:

### Step 1: Resolve the Initial Velocity
The initial velocity \( u \) can be resolved into its horizontal and vertical components:
- Horizontal component: \( u_x = u \cos \theta \)
- Vertical component: \( u_y = u \sin \theta \)

### Step 2: Write the Expression for Final Velocity
The final velocity \( \mathbf{v} \) of the projectile at time \( t \) can be expressed as:
\[ \mathbf{v} = u_x \hat{i} + (u_y - gt) \hat{j} \]
Substituting the components:
\[ \mathbf{v} = (u \cos \theta) \hat{i} + (u \sin \theta - gt) \hat{j} \]

### Step 3: Condition for Perpendicular Vectors
For the velocity vector \( \mathbf{v} \) to be perpendicular to the initial velocity vector \( \mathbf{u} \), their dot product must be zero:
\[ \mathbf{u} \cdot \mathbf{v} = 0 \]
Substituting the vectors:
\[ (u \cos \theta \hat{i} + u \sin \theta \hat{j}) \cdot ((u \cos \theta) \hat{i} + (u \sin \theta - gt) \hat{j}) = 0 \]

### Step 4: Expand the Dot Product
Calculating the dot product:
\[ (u \cos \theta)(u \cos \theta) + (u \sin \theta)(u \sin \theta - gt) = 0 \]
This simplifies to:
\[ u^2 \cos^2 \theta + u \sin \theta (u \sin \theta - gt) = 0 \]

### Step 5: Simplify the Equation
Expanding the second term:
\[ u^2 \cos^2 \theta + u^2 \sin^2 \theta - u g t \sin \theta = 0 \]
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ u^2 - u g t \sin \theta = 0 \]

### Step 6: Solve for Time \( t \)
Rearranging the equation gives:
\[ u g t \sin \theta = u^2 \]
Dividing both sides by \( u g \sin \theta \) (assuming \( u \neq 0 \) and \( \sin \theta \neq 0 \)):
\[ t = \frac{u}{g \sin \theta} \]

### Final Answer
Thus, the time when the velocity vector is perpendicular to the initial velocity vector is:
\[ t = \frac{u}{g \sin \theta} \]

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