Problem: Find the value of speed (km/hr) with which a projectile should be launched from the surface of the earth so as to reach a height h equal to 1/4th of the radius r of the earth.

Solution:

To solve this problem, we can use the principles of projectile motion and the gravitational force acting on the projectile.

1. Understanding the Problem:

- We need to find the speed at which the projectile should be launched from the surface of the earth to reach a height h.
- The height h is given as 1/4th of the radius r of the earth.

2. Analyzing the Situation:

- The projectile will follow a curved path due to the gravitational force acting on it.
- At the highest point of its trajectory, the projectile will have zero vertical velocity.
- The time taken to reach the highest point can be determined using the kinematic equation: vf = vi + at, where vf = 0 (final velocity), vi = initial velocity, a = acceleration (due to gravity), and t = time.
- The time taken to reach the highest point will be equal to the time taken to fall back to the surface of the earth.

3. Determining the Time of Flight:

- The time taken to reach the highest point can be calculated using the formula: t = (2 * vi) / g, where g is the acceleration due to gravity.
- The total time of flight will be twice the time taken to reach the highest point (as it takes the same amount of time to fall back to the surface).
- So, the total time of flight will be: T = 2 * [(2 * vi) / g] = (4 * vi) / g.

4. Finding the Required Speed:

- The required speed can be found by considering the vertical motion of the projectile.
- At the highest point, the vertical displacement will be equal to h.
- Using the equation of motion: s = vit + (1/2)at^2, where s = displacement, vi = initial velocity, t = time, and a = acceleration.
- At the highest point, the displacement is h, the initial velocity is vi, and the acceleration is -g (negative due to the direction).
- So, we can write h = vi * t - (1/2)g * t^2.
- Replacing t with (4 * vi) / g (time of flight), we get the equation: h = vi * [(4 * vi) / g] - (1/2)g * [((4 * vi) / g)^2].
- Simplifying the equation, we get: h = (4 * vi^2) / g - (8 * vi^2) / g.
- Rearranging the terms, we get: (8 * vi^2) / g = (4 * vi^2) / g - h.
- Simplifying further, we get: (4 * vi^2) / g = h.
- Rearranging the terms and substituting the value of h as (1/4)r (1/4th of the radius r of the earth), we get: vi^2 = (1/4)gr.
- Now, we can solve for vi (initial velocity) by taking