From the top of a tower of height 40m, a ball is projected upward with a speed of `20ms^(-1)` at an angle of elevation of `30^(@)`. Then the ratio of the total time taken by the ball to hit the ground to the time taken to ball come at same level as top of tower.

Options

A. `2:1`
B. `3:1`
C. `3:2`
D. `4:1`

Answer

To solve the problem step by step, we need to find the ratio of the total time taken by the ball to hit the ground to the time taken by the ball to come back to the same level as the top of the tower.

### Step 1: Resolve the initial velocity into components
The ball is projected with a speed of \(20 \, \text{m/s}\) at an angle of \(30^\circ\). We can resolve this velocity into horizontal and vertical components.

- **Vertical component** \(v_y = v \sin \theta = 20 \sin 30^\circ = 20 \times \frac{1}{2} = 10 \, \text{m/s}\)
- **Horizontal component** \(v_x = v \cos \theta = 20 \cos 30^\circ = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \, \text{m/s}\)

### Step 2: Calculate the time taken to reach the maximum height
The time taken to reach the maximum height can be calculated using the formula:
\[ t_{\text{up}} = \frac{v_y}{g} = \frac{10}{10} = 1 \, \text{s} \]

### Step 3: Calculate the maximum height reached
The maximum height \(h\) reached above the point of projection can be calculated using:
\[ h = v_y t_{\text{up}} - \frac{1}{2} g t_{\text{up}}^2 = 10 \times 1 - \frac{1}{2} \times 10 \times (1)^2 = 10 - 5 = 5 \, \text{m} \]
Thus, the total height from the ground is:
\[ H = 40 + 5 = 45 \, \text{m} \]

### Step 4: Calculate the total time of flight until it hits the ground
The total time of flight can be calculated using the formula:
\[ t_{\text{total}} = \frac{2v_y}{g} + t_{\text{fall}} \]
Where \(t_{\text{fall}}\) is the time taken to fall from the maximum height to the ground. We can use the equation of motion:
\[ H = \frac{1}{2} g t_{\text{fall}}^2 \Rightarrow 45 = \frac{1}{2} \times 10 \times t_{\text{fall}}^2 \Rightarrow 45 = 5 t_{\text{fall}}^2 \Rightarrow t_{\text{fall}}^2 = 9 \Rightarrow t_{\text{fall}} = 3 \, \text{s} \]
Thus, the total time of flight is:
\[ t_{\text{total}} = 1 + 3 = 4 \, \text{s} \]

### Step 5: Calculate the time taken to return to the same level as the top of the tower
The time taken to return to the same level as the top of the tower is simply the time taken to reach the maximum height:
\[ t_{\text{same level}} = 1 \, \text{s} \]

### Step 6: Calculate the ratio of the total time to hit the ground to the time taken to come back to the same level
The ratio is given by:
\[ \text{Ratio} = \frac{t_{\text{total}}}{t_{\text{same level}}} = \frac{4}{1} = 4:1 \]

### Final Answer
The ratio of the total time taken by the ball to hit the ground to the time taken for the ball to come back to the same level as the top of the tower is \(4:1\).

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