A stone is projected with a velocity `20sqrt2 m//s` at an angle of `45^(@)` to the horizontal.The average velocity of stone during its motion from starting point to its maximum height is

Options

A. `10sqrt5 m//s`
B. `20sqrt5 m//s`
C. `5sqrt5 m//s`
D. `20 m//s`

Explanation

To find the average velocity of the stone during its motion from the starting point to its maximum height, we can follow these steps:

### Step 1: Determine the initial velocity components
The stone is projected with a velocity of \( 20\sqrt{2} \, \text{m/s} \) at an angle of \( 45^\circ \) to the horizontal. We can break this velocity into its horizontal and vertical components using trigonometric functions.

- The horizontal component \( V_{x} \) is given by:
\[ V_{x} = V \cos(\theta) = 20\sqrt{2} \cos(45^\circ) = 20\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 20 \, \text{m/s} \]

- The vertical component \( V_{y} \) is given by:
\[ V_{y} = V \sin(\theta) = 20\sqrt{2} \sin(45^\circ) = 20\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 20 \, \text{m/s} \]

### Step 2: Calculate the time to reach maximum height
At maximum height, the vertical component of the velocity becomes zero. We can use the following kinematic equation:
\[ V_{y} = U_{y} - g t \]
Where:
- \( V_{y} = 0 \) (at maximum height)
- \( U_{y} = 20 \, \text{m/s} \)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( t \) is the time taken to reach maximum height.

Setting \( V_{y} = 0 \):
\[ 0 = 20 - 9.8 t \]
Solving for \( t \):
\[ 9.8 t = 20 \implies t = \frac{20}{9.8} \approx 2.04 \, \text{s} \]

### Step 3: Calculate the distance traveled in the vertical direction
The distance traveled in the vertical direction to reach maximum height can be calculated using the formula:
\[ s = U_{y} t - \frac{1}{2} g t^2 \]
Substituting the values:
\[ s = 20 \cdot 2.04 - \frac{1}{2} \cdot 9.8 \cdot (2.04)^2 \]
Calculating:
\[ s = 40.8 - \frac{1}{2} \cdot 9.8 \cdot 4.16 \approx 40.8 - 20.4 \approx 20.4 \, \text{m} \]

### Step 4: Calculate the average velocity
The average velocity \( V_{avg} \) during the time from the starting point to the maximum height can be calculated using the formula:
\[ V_{avg} = \frac{\text{Total displacement}}{\text{Total time}} \]
The total displacement in the vertical direction is \( 20.4 \, \text{m} \) and the total time is \( 2.04 \, \text{s} \).

Thus:
\[ V_{avg} = \frac{20.4}{2.04} \approx 10 \, \text{m/s} \]

### Final Step: Conclusion
The average velocity of the stone during its motion from the starting point to its maximum height is approximately \( 10 \, \text{m/s} \).

---