A stone is projected from the top of a tower with velocity `20ms^(-1)` making an angle `30^(@)` with the horizontal. If the total time of flight is 5s and `g=10ms^(-2)`

Options

A. the height of the tower is 75m
B. the maximum height of the stone from the ground is 80m
C. both of the above are true
D. none of the above is true

Answer

To solve the problem step by step, we will determine the height of the tower and the maximum height reached by the stone projected from the tower.

### Step 1: Break down the initial velocity into components
The stone is projected with a velocity of \( u = 20 \, \text{m/s} \) at an angle of \( \theta = 30^\circ \) with the horizontal. We can find the horizontal and vertical components of the initial velocity using trigonometric functions.

- **Horizontal component** (\( u_x \)):
\[ u_x = u \cos \theta = 20 \cos 30^\circ = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \, \text{m/s} \]

- **Vertical component** (\( u_y \)):
\[ u_y = u \sin \theta = 20 \sin 30^\circ = 20 \times \frac{1}{2} = 10 \, \text{m/s} \]

### Step 2: Use the time of flight to find the height of the tower
The total time of flight is given as \( T = 5 \, \text{s} \). We can use the second equation of motion to find the vertical displacement (height of the tower).

The equation is:
\[ s = u_y T + \frac{1}{2} a T^2 \]
where \( a = -g = -10 \, \text{m/s}^2 \) (acceleration due to gravity).

Substituting the values:
\[ s = (10 \, \text{m/s})(5 \, \text{s}) + \frac{1}{2}(-10 \, \text{m/s}^2)(5 \, \text{s})^2 \]
\[ s = 50 \, \text{m} - \frac{1}{2}(10)(25) \]
\[ s = 50 \, \text{m} - 125 \, \text{m} \]
\[ s = -75 \, \text{m} \]

The negative sign indicates that the stone has fallen 75 meters below the point of projection, which means the height of the tower is \( 75 \, \text{m} \).

### Step 3: Calculate the maximum height reached by the stone
To find the maximum height reached by the stone above the point of projection, we will use the following kinematic equation:
\[ v^2 = u^2 + 2as \]
At the maximum height, the final vertical velocity \( v = 0 \).

Substituting the values:
\[ 0 = (10 \, \text{m/s})^2 + 2(-10 \, \text{m/s}^2)s \]
\[ 0 = 100 - 20s \]
\[ 20s = 100 \]
\[ s = 5 \, \text{m} \]

### Step 4: Calculate the total height from the ground
The total height from the ground is the sum of the height of the tower and the maximum height reached by the stone:
\[ \text{Total height} = \text{Height of tower} + \text{Maximum height} \]
\[ \text{Total height} = 75 \, \text{m} + 5 \, \text{m} = 80 \, \text{m} \]

### Final Answers
- Height of the tower: \( 75 \, \text{m} \)
- Maximum height above the ground: \( 80 \, \text{m} \)

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