A projectile takes off with an initial velocity of 50m//s at an angle of `37^(@)` with horizontal, It is just able to clear two hurdles of height `25m` each, separated from each other by a distance `d` Calculate `d`.

Options

A. `160m`
B. `50m`
C. `320m`
D. `40m`

Explanation

To solve the problem of finding the distance \( d \) between two hurdles that a projectile just clears, we can break down the solution into several steps.

### Step-by-Step Solution:

1. **Identify the Components of Initial Velocity**:
The projectile is launched with an initial velocity \( u = 50 \, \text{m/s} \) at an angle \( \theta = 37^\circ \).
- The horizontal component of the velocity \( u_x \) is given by:
\[ u_x = u \cos(\theta) = 50 \cos(37^\circ) = 50 \times \frac{4}{5} = 40 \, \text{m/s} \]
- The vertical component of the velocity \( u_y \) is given by:
\[ u_y = u \sin(\theta) = 50 \sin(37^\circ) = 50 \times \frac{3}{5} = 30 \, \text{m/s} \]

2. **Use the Second Equation of Motion for Vertical Displacement**:
The height of the hurdles is \( h = 25 \, \text{m} \). We can use the second equation of motion to find the time \( t \) when the projectile reaches this height:
\[ h = u_y t - \frac{1}{2} g t^2 \]
where \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).
Substituting the values:
\[ 25 = 30t - \frac{1}{2} \cdot 10 \cdot t^2 \]
Simplifying gives:
\[ 25 = 30t - 5t^2 \]
Rearranging this into standard quadratic form:
\[ 5t^2 - 30t + 25 = 0 \]
Dividing the entire equation by 5:
\[ t^2 - 6t + 5 = 0 \]

3. **Solve the Quadratic Equation**:
We can factor this quadratic equation:
\[ (t - 1)(t - 5) = 0 \]
Thus, the solutions for \( t \) are:
\[ t_1 = 1 \, \text{s} \quad \text{and} \quad t_2 = 5 \, \text{s} \]

4. **Calculate the Distance \( d \)**:
The horizontal distance \( d \) between the two hurdles can be calculated using the horizontal velocity:
\[ d = u_x (t_2 - t_1) = 40 \, \text{m/s} \times (5 \, \text{s} - 1 \, \text{s}) = 40 \times 4 = 160 \, \text{m} \]

### Final Answer:
The distance \( d \) between the two hurdles is \( 160 \, \text{m} \).

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