Determine the Maximum Height a Ball Reaches When Thrown Vertically Upwards

In this article, we will solve a classic physics problem involving the vertical motion of a ball. Specifically, we'll calculate the maximum height that a ball reaches when thrown vertically upwards and returns to the ground after 4 seconds. Understanding these principles is not only important for physics enthusiasts but also has practical applications in fields such as sports science, engineering, and architecture. Let's dive into the detailed solution step-by-step.

Problem Statement and Assumptions

A ball is thrown vertically upwards from ground level and hits the ground after 4 seconds. We need to determine the maximum height it reached during its journey.

Step-by-Step Analysis

Step 1: Calculate the Time to Reach Maximum Height

Given that the ball is thrown vertically upwards and returns to the ground after 4 seconds, the time to reach the maximum height is half of this total time. Therefore, we can calculate the time to reach the maximum height as:

[mathjax] t_{text{up}} frac{4 , text{seconds}}{2} 2 , text{seconds}

Step 2: Use the Equation of Motion for Maximum Height

The maximum height ( h ) can be determined using the following equation:

[mathjax] h v_i t_{text{up}} - frac{1}{2} g t_{text{up}}^2

where:

( v_i ) is the initial velocity ( g ) is the acceleration due to gravity (approximately 9.81 m/s2) ( t_{text{up}} ) is the time taken to reach the maximum height (2 seconds)

Step 3: Determine the Initial Velocity

To find the initial velocity ( v_i ), we can use the fact that at the maximum height, the final velocity ( v_f ) is 0. The equation for this is:

[mathjax] v_f v_i - g t_{text{up}}

Setting ( v_f 0 ), we can solve for ( v_i ):

[mathjax] 0 v_i - 9.81 times 2 implies v_i 9.81 times 2 19.62 , text{m/s}

Step 4: Calculate the Maximum Height

Now, substituting ( v_i ) back into the maximum height equation:

[mathjax] begin{aligned} h 19.62 times 2 - frac{1}{2} times 9.81 times 2^2 39.24 - frac{1}{2} times 9.81 times 4 39.24 - 19.62 19.62 , text{meters}end{aligned}

Therefore, the maximum height reached by the ball is 19.62 meters.

Verification Using an Alternative Method

For verification, we can use a different set of equations and assumptions. Let's consider a different scenario where an object returns to the ground after 4 seconds from an initial height of 3.4 feet with an upward initial velocity. We will calculate the initial velocity and then the maximum height it can reach.

Step 1: Determine the Initial Velocity

Using the one-dimensional displacement equation:

[mathjax] s ut 1/2 a t^2

where:

( s -3.4 ft ) (displacement from initial height to ground) ( u ) is the initial velocity ( a -32.2 ft/s^2 ) (gravitational acceleration) ( t 0.8 s ) (time to hit the ground)

Substitute the known values:

[mathjax] -3.4 ft u times 0.8 s frac{1}{2} times (-32.2 , text{ft/s}^2) times (0.8 , text{s})^2

Calculating the terms:

[mathjax] -3.4 0.8u -10.304

Solving for ( u ):

[mathjax] 0.8u 10.304 3.4 implies u frac{10.304 3.4}{0.8} 8.63 , text{ft/s}

Step 2: Calculate the Maximum Height

At maximum height, the final velocity ( v_f ) is 0. Using the equation of motion:

[mathjax] v_f^2 u^2 2as

where:

( u 8.63 , text{ft/s} ) ( v_f 0 ) ( a -32.2 , text{ft/s}^2 ) ( s X - 3.4 , text{ft} ) (displacement from initial height to maximum height)

Substitute the values:

[mathjax] 0^2 8.63^2 2 times (-32.2) times (X - 3.4)

Calculate the terms:

[mathjax] begin{aligned} 0 74.4769 -64.4X - 218.96 -64.4X 293.4369 - 218.96 -64.4X 74.4769 X frac{74.4769}{64.4} X 4.5564736 , text{ft}end{aligned}

Therefore, the maximum height reached by the ball is approximately 4.6 feet.

Conclusion

In both scenarios, we have determined the maximum height reached by the ball. For the first scenario, the ball reached a height of 19.62 meters, while in the second scenario, the ball reached a height of approximately 4.6 feet. These calculations demonstrate the application of basic kinematic equations in solving real-world problems involving vertical motion.

Understanding these principles can be invaluable in many fields, including sports science, where the trajectory of a ball is crucial for improving performance, engineering, where it is essential for the design of structures, and even in architecture, where the influence of gravity in building design is a critical factor.

For more detailed information and to explore similar problems, please refer to educational resources or physics textbooks.