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. $2gH = n^2u^2$
• B. $ gH = (n-2)^2 u^2$
• C. $2gH = nu^2 (n-2)$
• D. $ gH = (n-2)u^{2}$

Answer 1

The Correct Option is C

To solve this problem, we need to determine the relationship between the height of the tower $H$, the initial speed $u$ of the particle, and the factor $n$ which relates the time taken by the particle to hit the ground to the time taken to reach its highest point.

Step-by-step Solution:

1. Time to reach the highest point:
   When a particle is thrown upwards, it reaches the highest point where its velocity becomes zero. The time \( t_1 \) to reach this highest point is given by:
   t_1 = \frac{u}{g}
   where \( g \) is the acceleration due to gravity.
2. Total time of flight:
   The particle takes \( n \times t_1 \) time to hit the ground, so the total time \( T \) is:
   T = n \cdot \frac{u}{g}
3. Calculating the time to hit the ground:
   The total time \( T \) is composed of the time going up \( t_1 \), the time coming down from the highest point to the initial throw height, and then the time to fall from that height to the ground.
   
   Using the equation of motion:
   s = ut - \frac{1}{2} g t^2
   where \( s \) is the distance, \( u \) is the initial velocity, \( t \) is the time, and \( g \) is the acceleration due to gravity.
4. Height relationship:
   The height \( H \) from which it falls after coming to zero velocity at the highest point can be calculated:
   Plug into the equation:
   H = -\frac{1}{2} g(n\frac{u}{g})^2 + u(n\frac{u}{g})
   Simplifying this and setting it equal to the given answer options:
   H = \frac{u^2}{2g} [n(n-2)]
5. Comparing and equating the final expression with the given options, we get:
   2gH = nu^2 (n-2)
   This concludes that the relation between \( H, u \), and \( n \) is indeed: 2gH = nu^2 (n-2).

Thus, the correct answer is: 2gH = nu^2 (n-2)