A particle is projected with a speed `10sqrt(2)` making an angle `45^(@)` with the horizontal . Neglect the effect of air friction. Then after 1 second of projection . Take `g=10m//s^(2)`.

Options

A. the height of the aprticle above the point of projection is `5m`.
B. the height of the particle above the point of projection is `10m.`
C. the horizontal distance of the particle from the point of projection is `5m`.
D. the horizontal distance of the particle from the point of projection is `15m`.

Explanation

To solve the problem step by step, we will break down the motion of the particle into its horizontal and vertical components and use the relevant equations of motion.

### Given Data:
- Initial speed, \( u = 10\sqrt{2} \, \text{m/s} \)
- Angle of projection, \( \theta = 45^\circ \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
- Time of projection, \( t = 1 \, \text{s} \)

### Step 1: Calculate the horizontal and vertical components of the initial velocity.
The initial velocity can be resolved into horizontal and vertical components using trigonometric functions.

1. **Horizontal component (\( u_x \))**:
\[ u_x = u \cos(\theta) = 10\sqrt{2} \cos(45^\circ) = 10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10 \, \text{m/s} \]

2. **Vertical component (\( u_y \))**:
\[ u_y = u \sin(\theta) = 10\sqrt{2} \sin(45^\circ) = 10\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 10 \, \text{m/s} \]

### Step 2: Calculate the height of the particle after 1 second.
We will use the vertical motion equation to find the height \( S_y \) after 1 second.

The formula for vertical displacement is:
\[ S_y = u_y t + \frac{1}{2}(-g)t^2 \]
Substituting the known values:
\[ S_y = 10 \cdot 1 + \frac{1}{2}(-10)(1^2) \]
\[ S_y = 10 - 5 = 5 \, \text{m} \]

### Step 3: Calculate the horizontal distance traveled after 1 second.
Since there is no horizontal acceleration, the horizontal distance \( S_x \) can be calculated as:
\[ S_x = u_x t \]
Substituting the known values:
\[ S_x = 10 \cdot 1 = 10 \, \text{m} \]

### Final Results:
- Height of the particle above the point of projection after 1 second: **5 meters**
- Horizontal distance from the point of projection after 1 second: **10 meters**

### Summary of Results:
- Height after 1 second: **5 m**
- Horizontal distance after 1 second: **10 m**