Question

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 relation between \( H \), \( u \) and \( n \) is

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

Hint:

When dealing with vertical motion under gravity from a height, choosing a consistent sign convention (e.g., upward displacement and velocity as positive, downward as negative) prevents sign errors.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
A particle is projected upwards from the top of a tower of height \( H \) with velocity \( u \). We need to find the relationship between \( H \), \( u \), and the factor \( n \), where the total time of flight to the ground is \( n \) times the time to reach the highest point.

Step 2: Key Formula or Approach:
- Time to reach the highest point \( t_1 \) is obtained using \( v = u - gt_1 \), where \( v = 0 \).
- Total time of flight \( t_2 = n \cdot t_1 \).
- Use the second equation of motion for vertical displacement from the top of the tower:
\[ s = ut - \frac{1}{2}gt^2 \]

Step 3: Detailed Explanation:
1. Finding time to the highest point (\( t_1 \)):
At the maximum height, the final velocity is zero:
\[ 0 = u - gt_1 \implies t_1 = \frac{u}{g} \]
2. Finding total time of flight (\( t_2 \)):
Given that the total time is \( n \) times \( t_1 \):
\[ t_2 = n t_1 = \frac{nu}{g} \]
3. Applying the displacement equation:
Taking the point of projection as the origin and upward direction as positive:
- Displacement, \( s = -H \)
- Initial velocity, \( u_i = +u \)
- Acceleration, \( a = -g \)
- Time, \( t = t_2 = \frac{nu}{g} \)
Substituting these into the displacement equation:
\[ -H = u\left(\frac{nu}{g}\right) - \frac{1}{2}g\left(\frac{nu}{g}\right)^2 \]
\[ -H = \frac{nu^2}{g} - \frac{1}{2}\frac{n^2u^2}{g} \]
Multiply the entire equation by \( -2g \) to simplify:
\[ 2gH = -2nu^2 + n^2u^2 \]
\[ 2gH = n^2u^2 - 2nu^2 \]
Factoring out \( nu^2 \):
\[ 2gH = nu^2(n - 2) \]

Step 4: Final Answer:
The relation is \( 2gH = nu^2(n - 2) \).