Derive the expression for height of a tower when a body is projected vertically upwards from the tower

To derive the expression for the height of a tower when a body is projected vertically upwards from it, we need to consider the motion of the body under the influence of gravity. Let's break this down step by step.

Understanding the Motion

When a body is projected upwards from a height (the tower), it will first ascend until it reaches its maximum height, and then it will descend back to the ground. The key factors to consider are the initial velocity of the projection, the height of the tower, and the acceleration due to gravity.

Key Variables

• h: height of the tower
• u: initial velocity of the body when projected upwards
• g: acceleration due to gravity (approximately 9.81 m/s², acting downwards)
• t: time taken to reach the maximum height
• H: maximum height reached above the ground

Equations of Motion

We can use the equations of motion to analyze the upward and downward journey of the body. The first step is to determine the time taken to reach the maximum height.

Time to Reach Maximum Height

At the maximum height, the final velocity (v) becomes zero. Using the first equation of motion:

v = u - gt

Setting v = 0, we have:

0 = u - gt

From this, we can solve for time (t):

t = u/g

Maximum Height Above the Tower

Next, we can find the maximum height (H) reached above the tower using the second equation of motion:

H = ut - (1/2)gt²

Substituting the value of t we found earlier:

H = u(u/g) - (1/2)g(u/g)²

This simplifies to:

H = (u²/g) - (1/2)(u²/g)

Combining the terms gives:

H = (u²/g) / 2

Total Height Above Ground

The total height (H_total) from the ground when the body reaches its maximum height is the sum of the height of the tower (h) and the maximum height (H) reached above the tower:

H_total = h + H

Substituting the expression for H:

H_total = h + (u²/2g)

Final Expression

Thus, the expression for the height of the tower when a body is projected vertically upwards from it is:

h = H_total - (u²/2g)

Example for Clarity

Let's say a body is projected upwards with an initial velocity of 20 m/s from a tower. To find the height of the tower, we first calculate the maximum height reached above the tower:

H = (20²)/(2 * 9.81) ≈ 20.4 m

If the total height from the ground is 50 m, we can find the height of the tower:

h = 50 - 20.4 ≈ 29.6 m

This example illustrates how to apply the derived formula in a practical scenario. Understanding these concepts helps in grasping the principles of projectile motion and the effects of gravity on objects in motion.