A ball is thrown at a speed of 40 m/s at an angle of `60^0` with the horizontal. Find a. the maximum height reached and b. the range of te ball. Take `g=10 m/s^2`.

Explanation

To solve the problem of a ball thrown at a speed of 40 m/s at an angle of 60 degrees with the horizontal, we will break it down into two parts: finding the maximum height reached by the ball and finding the range of the ball.

### Part (a): Maximum Height Reached

1. **Identify the initial velocity components**:
- The initial speed \( U = 40 \, \text{m/s} \).
- The angle of projection \( \theta = 60^\circ \).
- The vertical component of the initial velocity \( U_y = U \sin(\theta) = 40 \sin(60^\circ) \).
- Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), we have:
\[ U_y = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3} \, \text{m/s} \]

2. **Use the kinematic equation to find the maximum height**:
- At maximum height, the final vertical velocity \( V_y = 0 \).
- We can use the equation:
\[ V_y^2 = U_y^2 - 2gH \]
- Rearranging gives:
\[ H = \frac{U_y^2}{2g} \]
- Substituting the values:
\[ H = \frac{(20\sqrt{3})^2}{2 \times 10} = \frac{1200}{20} = 60 \, \text{m} \]

### Part (b): Range of the Ball

1. **Calculate the time of flight**:
- The time to reach maximum height \( t_{up} \) can be found using:
\[ V_y = U_y - gt \]
- Setting \( V_y = 0 \):
\[ 0 = 20\sqrt{3} - 10t_{up} \]
- Solving for \( t_{up} \):
\[ t_{up} = \frac{20\sqrt{3}}{10} = 2\sqrt{3} \, \text{s} \]
- The total time of flight \( T \) is twice the time to reach maximum height:
\[ T = 2t_{up} = 4\sqrt{3} \, \text{s} \]

2. **Calculate the horizontal component of the initial velocity**:
- The horizontal component \( U_x = U \cos(\theta) = 40 \cos(60^\circ) \).
- Since \( \cos(60^\circ) = \frac{1}{2} \):
\[ U_x = 40 \times \frac{1}{2} = 20 \, \text{m/s} \]

3. **Calculate the range**:
- The range \( R \) can be calculated using:
\[ R = U_x \times T \]
- Substituting the values:
\[ R = 20 \times 4\sqrt{3} = 80\sqrt{3} \, \text{m} \]

### Final Answers
- Maximum Height \( H = 60 \, \text{m} \)
- Range \( R = 80\sqrt{3} \, \text{m} \)