A stone is projected from the ground with velocity `50(m)/(s)` at an angle of `30^@`. It crosses a wall after 3 sec. How far beyond the wall the stone will strike the ground `(g=10(m)/(sec^2)`?

Options

A. 90.2 m
B. 12.5 m
C. 86.5 m
D. 216.25 m

Answer

To solve the problem step by step, we will follow the principles of projectile motion.

### Step 1: Identify the components of the initial velocity
The stone is projected with an initial velocity \( u = 50 \, \text{m/s} \) at an angle \( \theta = 30^\circ \).

The horizontal and vertical components of the initial velocity can be calculated as:
- Horizontal component \( u_x = u \cos \theta = 50 \cos(30^\circ) \)
- Vertical component \( u_y = u \sin \theta = 50 \sin(30^\circ) \)

Using the values:
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(30^\circ) = \frac{1}{2} \)

Calculating:
- \( u_x = 50 \cdot \frac{\sqrt{3}}{2} = 25\sqrt{3} \, \text{m/s} \)
- \( u_y = 50 \cdot \frac{1}{2} = 25 \, \text{m/s} \)

### Step 2: Calculate the total time of flight
The total time of flight \( T \) for a projectile is given by:
\[ T = \frac{2u_y}{g} \]
where \( g = 10 \, \text{m/s}^2 \).

Substituting the values:
\[ T = \frac{2 \cdot 25}{10} = 5 \, \text{s} \]

### Step 3: Determine the time in the air after crossing the wall
The stone crosses the wall after \( t = 3 \, \text{s} \). Therefore, the time remaining in the air after crossing the wall is:
\[ t_{\text{remaining}} = T - t = 5 - 3 = 2 \, \text{s} \]

### Step 4: Calculate the horizontal distance traveled after crossing the wall
The horizontal distance \( d \) traveled after crossing the wall can be calculated using the horizontal component of the velocity and the remaining time:
\[ d = u_x \cdot t_{\text{remaining}} \]
Substituting the values:
\[ d = 25\sqrt{3} \cdot 2 = 50\sqrt{3} \, \text{m} \]

### Step 5: Calculate the numerical value of the distance
Using \( \sqrt{3} \approx 1.732 \):
\[ d \approx 50 \cdot 1.732 \approx 86.6 \, \text{m} \]

### Final Answer
The stone will strike the ground approximately **86.6 meters** beyond the wall.

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