A particle is projected with a speed of 10 m/s at an angle `37^(@)` with the vertical. Find (i) time of flight (ii) maximum height above ground (iii) horizontal range.

Answer

To solve the problem step by step, we will find the time of flight, maximum height above the ground, and horizontal range of the particle projected at a speed of 10 m/s at an angle of 37 degrees with the vertical.

### Step 1: Determine the angle with respect to the horizontal
Since the angle given is with respect to the vertical, we can find the angle with respect to the horizontal:
\[ \theta = 90^\circ - 37^\circ = 53^\circ \]

### Step 2: Calculate the time of flight
The formula for the time of flight \( T \) for a projectile is given by:
\[ T = \frac{2 V_0 \sin \theta}{g} \]
Where:
- \( V_0 = 10 \, \text{m/s} \) (initial speed)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( \theta = 53^\circ \)

Now, substituting the values:
\[ T = \frac{2 \times 10 \times \sin(53^\circ)}{10} \]
Using \( \sin(53^\circ) \approx \frac{4}{5} \):
\[ T = \frac{2 \times 10 \times \frac{4}{5}}{10} = \frac{2 \times 8}{10} = 1.6 \, \text{s} \]

### Step 3: Calculate the maximum height above ground
The formula for maximum height \( H \) is given by:
\[ H = \frac{V_0^2 \sin^2 \theta}{2g} \]
Substituting the values:
\[ H = \frac{10^2 \times \left(\frac{4}{5}\right)^2}{2 \times 10} \]
Calculating:
\[ H = \frac{100 \times \frac{16}{25}}{20} = \frac{1600}{500} = 3.2 \, \text{m} \]

### Step 4: Calculate the horizontal range
The formula for horizontal range \( R \) is given by:
\[ R = \frac{V_0^2 \sin(2\theta)}{g} \]
Using \( \sin(2\theta) = 2 \sin \theta \cos \theta \):
\[ \sin(2\theta) = 2 \times \sin(53^\circ) \times \cos(53^\circ) \]
Where \( \cos(53^\circ) \approx \frac{3}{5} \):
\[ R = \frac{10^2 \times 2 \times \frac{4}{5} \times \frac{3}{5}}{10} \]
Calculating:
\[ R = \frac{100 \times \frac{24}{25}}{10} = \frac{2400}{250} = 9.6 \, \text{m} \]

### Summary of Results
1. Time of Flight: \( T = 1.6 \, \text{s} \)
2. Maximum Height: \( H = 3.2 \, \text{m} \)
3. Horizontal Range: \( R = 9.6 \, \text{m} \)