Problem Statement: Two stones are projected from the top of a tower 100m high each with the velocity of 20m/s. One is projected vertically upwards and another vertically downwards. Calculate the time each stone takes to reach the ground and the velocity with which it strikes the ground.

Solution:

Given:

• Height of the tower (h) = 100m


• Initial velocity (u) = 20m/s

1. Time taken for the stone projected vertically upwards to reach the ground:

When a stone is projected vertically upwards, the initial velocity is in the upward direction and gravity acts in the downward direction. Hence, the acceleration (a) is -9.8m/s² (negative sign indicates that the acceleration is in the downward direction).

Using the equation of motion, we can find the time taken for the stone to reach the ground:

h = ut + 1/2at²

Where h is the height of the tower, u is the initial velocity, a is the acceleration and t is the time taken.

Substituting the given values, we get:

100 = 20t + 1/2(-9.8)t²

Simplifying the equation, we get:

4.9t² - 20t + 100 = 0

Using the quadratic formula, we get:

t = 4.04s (approx.)

Hence, the time taken for the stone projected vertically upwards to reach the ground is approximately 4.04 seconds.

2. Time taken for the stone projected vertically downwards to reach the ground:

When a stone is projected vertically downwards, the initial velocity is in the downward direction and gravity acts in the downward direction. Hence, the acceleration (a) is 9.8m/s² (positive sign indicates that the acceleration is in the downward direction).

Using the same equation of motion, we can find the time taken for the stone to reach the ground:

h = ut + 1/2at²

Where h is the height of the tower, u is the initial velocity, a is the acceleration and t is the time taken.

Substituting the given values, we get:

100 = -20t + 1/2(9.8)t²

Simplifying the equation, we get:

4.9t² + 20t + 100 = 0

Using the quadratic formula, we get:

t = 2.04s (approx.)

Hence, the time taken for the stone projected vertically downwards to reach the ground is approximately 2.04 seconds.

3. Velocity with which each stone strikes the ground:

When a stone falls freely under the influence of gravity, its final velocity (v) can be calculated using the following equation:

v² = u² + 2as

Where u is the initial velocity, a is the acceleration due to gravity and s is the distance fallen.

For the stone projected vertically upwards:

Since the stone is projected upwards with an initial velocity of 20m/s, its final velocity when it strikes the ground will be in the downward direction and its magnitude will be equal to the initial velocity. Hence, we can use the above equation to find the final