Calculating the Height of a Tower Using Projectile Motion

In this article, we will explore the application of the equations of motion under uniform acceleration due to gravity to determine the height of a tower. We will use a practical example of a ball being thrown upwards to illustrate the process. By the end of this article, you will understand how to apply these principles to similar problems.

Problem Statement

A ball is thrown upwards with an initial velocity of 19.6 m/s. It returns to the ground in 5 seconds. We need to find the height of the tower from which the ball was thrown.

Concepts Involved

To solve this problem, we will use the following concepts:

Equations of Motion: These are fundamental in solving problems of motion under uniform acceleration. The equations we will use are: Final velocity v initial velocity u acceleration a t Distance s initial velocity u t (1/2) a t^2 Final velocity v^2 initial velocity u^2 2 a s

Solution

Let's break down the problem step by step.

Step 1: Calculate the Maximum Height

At the maximum height, the ball comes to a stop, so the final velocity v is 0 m/s.

Using the equation: V u at

Setting v 0,

0 19.6 (-9.8)t

Solving for t,

9.8t 19.6 t 19.6 / 9.8 2 seconds

Step 2: Calculate the Height Reached in This Time

The ball takes 2 seconds to reach the maximum height. We use the distance equation to calculate the height:

s ut (1/2) at^2

where s is the height reached, u 19.6 m/s, a -9.8 m/s^2, and t 2 s.

s 19.6(2) (1/2)(-9.8)(2^2)

s 39.2 - 19.6

s 19.6 meters

Step 3: Calculate the Total Height of the Tower

The total time of flight is 5 seconds. The ball takes 2 seconds to reach the maximum height and 3 seconds to fall back down. The height fallen during the last 3 seconds can be calculated using the distance equation again:

s ut (1/2) at^2

where u 0 m/s, a 9.8 m/s^2, and t 3 s.

s 0(3) (1/2)(9.8)(3^2)

s (1/2)88.2

s 44.1 meters

Therefore, the total height of the tower is the sum of the height reached and the height fallen:

Total height 19.6 m 44.1 m 63.7 meters

Conclusion

The height of the tower is 63.7 meters. This is achieved by applying the equations of motion and understanding the physical principles involved in projectile motion. By breaking down the problem into smaller steps, we can accurately determine the height of the tower.

Additional Notes

It's important to note that the formula (frac{1}{2} g t^2) should not be used if the initial velocity is not zero, as in this case. The formula takes into account only the distance fallen due to gravity, ignoring the initial velocity. In this problem, the ball has an initial velocity of 19.6 m/s, so we need to account for it in our calculations.

To summarize, the height of the tower can be calculated by considering both the height reached during the upward motion and the distance fallen during the downward motion.