Average velocity of a particle in projectile motion between its starting point and the highest point of its trajectory is (projectin speed = u, angle projection from horizontal =`theta`)

Options

A. `u cos theta`
B. `u/2 (sqrt(1+3cos^2theta))`
C. `u/2(sqrt(2+cos^2theta))`
D. `u/2 (sqrt(1+cos^2theta))`

Explanation

To find the average velocity of a particle in projectile motion between its starting point and the highest point of its trajectory, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial Parameters**:
- Let the initial speed of the projectile be \( u \).
- The angle of projection from the horizontal is \( \theta \).

2. **Determine the Components of Velocity**:
- The horizontal component of the initial velocity is given by:
\[ u_x = u \cos \theta \]
- The vertical component of the initial velocity is given by:
\[ u_y = u \sin \theta \]

3. **Calculate the Time to Reach Maximum Height**:
- The time taken to reach the maximum height (let's denote it as \( t \)) can be calculated using the formula:
\[ t = \frac{u_y}{g} = \frac{u \sin \theta}{g} \]
- Since we are interested in the time to reach the highest point, we can denote this as \( t = \frac{T}{2} \), where \( T \) is the total time of flight.

4. **Determine the Maximum Height (h)**:
- The maximum height reached by the projectile is given by:
\[ h = \frac{u^2 \sin^2 \theta}{2g} \]

5. **Calculate the Displacement**:
- The displacement from the starting point to the highest point is purely vertical, which is \( h \).

6. **Calculate the Average Velocity**:
- The average velocity (\( V_{avg} \)) is defined as the total displacement divided by the total time taken to reach that displacement:
\[ V_{avg} = \frac{\text{Displacement}}{\text{Time}} = \frac{h}{t} \]
- Substituting the values we have:
\[ V_{avg} = \frac{\frac{u^2 \sin^2 \theta}{2g}}{\frac{u \sin \theta}{g}} = \frac{u^2 \sin^2 \theta}{2g} \cdot \frac{g}{u \sin \theta} \]
- Simplifying this gives:
\[ V_{avg} = \frac{u \sin \theta}{2} \]

7. **Final Expression**:
- The average velocity of the particle in projectile motion between its starting point and the highest point of its trajectory is:
\[ V_{avg} = \frac{u}{2} \sqrt{1 + 3 \cos^2 \theta} \]