A train is moving on a track at `30ms^-1`. A ball is thrown from it perpendicular to the direction of motion with `30 ms^-1` at `45^@` from horizontal. Find the distance of ball from the point of projection on train to the point where it strikes the ground.

Options

A. 90 m
B. `90(sqrt3)` m
C. 60 m
D. `60 (sqrt3)`m

Explanation

To solve the problem, we need to find the distance from the point of projection on the train to the point where the ball strikes the ground. The ball is thrown at an angle of 45 degrees from the horizontal with a speed of 30 m/s, while the train is moving horizontally at a speed of 30 m/s.

### Step-by-Step Solution:

1. **Determine the Components of the Ball's Velocity:**
The ball is thrown at an angle of 45 degrees with a speed of 30 m/s. We can resolve this velocity into horizontal and vertical components using trigonometric functions:
- Horizontal component (Vx) = V * cos(θ) = 30 * cos(45°) = 30 * (1/√2) = 30/√2 m/s
- Vertical component (Vy) = V * sin(θ) = 30 * sin(45°) = 30 * (1/√2) = 30/√2 m/s

2. **Calculate the Time of Flight (t):**
The time of flight can be calculated using the formula for vertical motion. The ball will rise and then fall back to the ground. The time to reach the maximum height is given by:
\[ t_{up} = \frac{Vy}{g} = \frac{30/√2}{9.81} \approx 2.14 \text{ seconds} \]
Since the time to go up is equal to the time to come down, the total time of flight (t) is:
\[ t = 2 * t_{up} \approx 2 * 2.14 \approx 4.28 \text{ seconds} \]

3. **Calculate the Horizontal Distance Covered by the Ball:**
The horizontal distance (D) covered by the ball while it is in the air can be calculated using the horizontal component of the velocity and the total time of flight:
\[ D = Vx * t = \left(\frac{30}{√2}\right) * 4.28 \approx 30 * 4.28 / 1.414 \approx 90.0 \text{ meters} \]

4. **Calculate the Distance Covered by the Train:**
The train is also moving while the ball is in the air. The distance covered by the train during the time of flight is:
\[ D_{train} = V_{train} * t = 30 * 4.28 \approx 128.4 \text{ meters} \]

5. **Calculate the Total Distance from the Point of Projection:**
The total distance from the point of projection on the train to the point where the ball strikes the ground is the sum of the distance covered by the train and the horizontal distance covered by the ball:
\[ D_{total} = D_{train} + D = 128.4 + 90.0 \approx 218.4 \text{ meters} \]

### Final Answer:
The distance of the ball from the point of projection on the train to the point where it strikes the ground is approximately **218.4 meters**.