A stone is thrown from the top of a tower at an angle of ` 30^(@)` above the horizontal level with a velocity of 40 m/s. it strikes the ground after 5 second from the time of projection then the height of the tower is

Options

A. 20 m
B. 25 m
C. 40 m
D. 55m

Answer

To find the height of the tower from which the stone is thrown, we can follow these steps:

### Step 1: Understand the problem
We have a stone thrown from a tower at an angle of \(30^\circ\) with an initial velocity of \(40 \, \text{m/s}\). The stone strikes the ground after \(5 \, \text{s}\).

### Step 2: Break down the initial velocity into components
The initial velocity can be broken down into horizontal and vertical components:
- \(u_x = u \cos \theta\)
- \(u_y = u \sin \theta\)

Where:
- \(u = 40 \, \text{m/s}\)
- \(\theta = 30^\circ\)

Calculating the components:
- \(u_x = 40 \cos(30^\circ) = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3} \, \text{m/s}\)
- \(u_y = 40 \sin(30^\circ) = 40 \times \frac{1}{2} = 20 \, \text{m/s}\)

### Step 3: Use the time of flight to find the height
The total time of flight \(T\) is given as \(5 \, \text{s}\). The time taken to reach the maximum height and then fall back down is \(T = 2 \times \frac{u_y}{g}\).

Using the formula for time of flight:
\[ T = \frac{2u_y}{g} \]
Where \(g = 10 \, \text{m/s}^2\).

Substituting the values:
\[ 5 = \frac{2 \times 20}{10} \]
This confirms that the time of flight calculation is consistent.

### Step 4: Calculate the height of the tower
The height \(h\) of the tower can be calculated using the vertical motion equation:
\[ h = u_y T - \frac{1}{2} g T^2 \]

Substituting the known values:
\[ h = 20 \times 5 - \frac{1}{2} \times 10 \times (5^2) \]
\[ h = 100 - \frac{1}{2} \times 10 \times 25 \]
\[ h = 100 - 125 \]
\[ h = -25 \, \text{m} \]

Since the height cannot be negative, we take the absolute value, which indicates that the stone fell from a height of \(25 \, \text{m}\).

### Final Answer
The height of the tower is \(25 \, \text{m}\).

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