From a tower of height `H`, a particle is thrown vertically upwards with a speed `u` . The time taken by the particle , to hit the ground , is ` n` times that taken by it to reach the highest point of its path. The relative between ` H, u and n` is :

Options

A. ` 2gH = n^(2)u^(2)`
B. ` gH = (n-2)^(2)u^(2)`
C. ` 2gH = nu^(2)(n-2)`
D. ` gH = (n- 2)u^(2)`

Explanation

To solve the problem, we need to establish the relationship between the height of the tower \( H \), the initial speed \( u \), and the time factor \( n \). Here’s a step-by-step breakdown of the solution:

### Step 1: Determine Time to Reach Maximum Height
When a particle is thrown vertically upwards with an initial speed \( u \), it will reach its maximum height when its velocity becomes zero. Using the equation of motion:
\[ v = u - gt \]
At maximum height, \( v = 0 \):
\[ 0 = u - gt \implies t = \frac{u}{g} \]
Thus, the time taken to reach the maximum height \( t \) is:
\[ t = \frac{u}{g} \]

### Step 2: Determine Total Time to Hit the Ground
According to the problem, the total time \( T \) taken to hit the ground is \( n \) times the time taken to reach the maximum height:
\[ T = n \cdot t = n \cdot \frac{u}{g} \]

### Step 3: Calculate Displacement During Total Time
The total displacement when the particle hits the ground from the height \( H \) is given by the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
Here, the displacement \( s \) is equal to \( -H \) (since it falls down), \( u \) is the initial speed, \( a = -g \) (acceleration due to gravity), and \( t = T \):
\[ -H = uT - \frac{1}{2} g T^2 \]

### Step 4: Substitute for \( T \)
Substituting \( T = n \cdot \frac{u}{g} \) into the displacement equation:
\[ -H = u \left(n \cdot \frac{u}{g}\right) - \frac{1}{2} g \left(n \cdot \frac{u}{g}\right)^2 \]
This simplifies to:
\[ -H = \frac{nu^2}{g} - \frac{1}{2} g \cdot \frac{n^2 u^2}{g^2} \]
\[ -H = \frac{nu^2}{g} - \frac{n^2 u^2}{2g} \]

### Step 5: Combine Terms
Now, we can combine the terms on the right side:
\[ -H = \frac{2nu^2 - n^2 u^2}{2g} \]
\[ -H = \frac{u^2 (2n - n^2)}{2g} \]

### Step 6: Rearranging for \( H \)
Multiplying both sides by -1:
\[ H = -\frac{u^2 (2n - n^2)}{2g} \]

### Step 7: Final Relation
Thus, the relationship between \( H \), \( u \), and \( n \) can be expressed as:
\[ H = \frac{u^2 (n^2 - 2n)}{2g} \]