A projectile is projected with speed u at an angle `theta` with the horizontal . The average velocity of the projectile between the instants it crosses the same level is

Options

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

Answer

To find the average velocity of a projectile between the instants it crosses the same level, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Average Velocity**:
Average velocity is defined as the total displacement divided by the total time taken. Mathematically, it can be expressed as:
\[ \text{Average Velocity} = \frac{\text{Change in Displacement}}{\text{Time Taken}} \]

2. **Identify the Motion of the Projectile**:
When a projectile is launched at an angle \(\theta\) with an initial speed \(u\), it has both horizontal and vertical components of velocity. The horizontal component of the initial velocity is given by:
\[ u_x = u \cos \theta \]
The vertical component is:
\[ u_y = u \sin \theta \]

3. **Determine the Displacement**:
When the projectile crosses the same level (the horizontal plane from which it was projected), the vertical displacement is zero. However, it travels a certain horizontal distance during its flight.

4. **Calculate the Time of Flight**:
The total time of flight \(T\) for a projectile is given by:
\[ T = \frac{2u \sin \theta}{g} \]
where \(g\) is the acceleration due to gravity.

5. **Calculate the Horizontal Displacement**:
The horizontal displacement \(R\) (range) can be calculated as:
\[ R = u_x \cdot T = (u \cos \theta) \cdot \left(\frac{2u \sin \theta}{g}\right) = \frac{2u^2 \sin \theta \cos \theta}{g} \]

6. **Calculate the Average Velocity**:
Since the vertical displacement is zero, the average velocity in the horizontal direction is simply the horizontal component of the initial velocity:
\[ \text{Average Velocity} = \frac{\text{Horizontal Displacement}}{\text{Time of Flight}} = \frac{R}{T} \]
However, since we are only interested in the average velocity between the two points where the projectile crosses the same level, we can directly use the horizontal component:
\[ \text{Average Velocity} = u \cos \theta \]

### Final Answer:
The average velocity of the projectile between the instants it crosses the same level is:
\[ \text{Average Velocity} = u \cos \theta \]