What is the average velocity of a particle projected from the ground with speed u at an angle alpha with horizontal over a time interval from beginning till it strikes the ground again?

Explanation

To find the average velocity of a particle projected from the ground with speed \( u \) at an angle \( \alpha \) with the horizontal until it strikes the ground again, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Average Velocity**:
The average velocity (\( V_{avg} \)) is defined as the total displacement divided by the total time taken. Mathematically, it can be expressed as:
\[ V_{avg} = \frac{\text{Total Displacement}}{\text{Total Time}} \]

2. **Identifying the Total Displacement**:
When the particle is projected and returns to the ground, the total displacement is the straight-line distance from the starting point to the endpoint. Since the particle returns to the same horizontal level (ground), the vertical displacement is zero. Therefore, the total displacement is equal to the horizontal range (\( R \)) of the projectile.

3. **Calculating the Horizontal Range**:
The horizontal range (\( R \)) of a projectile launched with speed \( u \) at an angle \( \alpha \) is given by the formula:
\[ R = \frac{u^2 \sin(2\alpha)}{g} \]

4. **Calculating the Total Time of Flight**:
The total time of flight (\( T \)) for the projectile is given by:
\[ T = \frac{2u \sin(\alpha)}{g} \]

5. **Substituting into the Average Velocity Formula**:
Now, substituting the values of total displacement and total time into the average velocity formula:
\[ V_{avg} = \frac{R}{T} = \frac{\frac{u^2 \sin(2\alpha)}{g}}{\frac{2u \sin(\alpha)}{g}} \]

6. **Simplifying the Expression**:
By simplifying the above expression, we can cancel \( g \) from the numerator and denominator:
\[ V_{avg} = \frac{u^2 \sin(2\alpha)}{2u \sin(\alpha)} = \frac{u \sin(2\alpha)}{2 \sin(\alpha)} \]
Using the identity \( \sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha) \):
\[ V_{avg} = \frac{u \cdot 2 \sin(\alpha) \cos(\alpha)}{2 \sin(\alpha)} = u \cos(\alpha) \]

7. **Final Result**:
Thus, the average velocity of the particle from the point of projection until it strikes the ground again is:
\[ V_{avg} = u \cos(\alpha) \]