The Physics of Vertical Motion

Oct 13, 2025 by windailys.com 73 views

Introduction

When a ball is thrown vertically upward, it experiences a downward acceleration due to gravity, which causes its velocity to decrease and its height to increase until it reaches its maximum height. After that, the ball begins to fall back down, and its velocity becomes more negative. In this article, we will explore the motion of a ball thrown vertically upward from a height of 2 m above the ground with an initial velocity of $$+15 , \text{m/s}$$. We will analyze the possible height and velocity pairs of the ball at some point in time.

The Physics of Vertical Motion

When an object is thrown vertically upward, it is subject to the force of gravity, which is a downward force. The acceleration due to gravity is denoted by the symbol $$g$$ and is equal to $$9.8 , \text{m/s}^2$$. The motion of the ball can be described using the equations of motion, which relate the position, velocity, and acceleration of the ball.

Equations of Motion

The equations of motion for an object under constant acceleration are:

$s = s_0 + v_0t + \frac{1}{2}at^2$
$v = v_0 + at$
$a = \frac{v - v_0}{t}$

where $$s$$ is the final position, $$s_0$$ is the initial position, $$v$$ is the final velocity, $$v_0$$ is the initial velocity, $$a$$ is the acceleration, and $$t$$ is the time.

Applying the Equations of Motion

We can apply the equations of motion to the ball thrown vertically upward. Let's assume that the ball is thrown from a height of 2 m above the ground with an initial velocity of $$+15 , \text{m/s}$$. We want to find the possible height and velocity pairs of the ball at some point in time.

Possible Height and Velocity Pairs

Using the equations of motion, we can find the height and velocity of the ball at any point in time. Let's consider the following pairs:

Pair 1: At $$t = 0.5 , \text{s}$$, the height of the ball is $$s = 2 + 15(0.5) - \frac{1}{2}(9.8)(0.5)^2 = 8.55 , \text{m}$$, and the velocity of the ball is $$v = 15 - 9.8(0.5) = 10.9 , \text{m/s}$$.
Pair 2: At $$t = 1 , \text{s}$$, the height of the ball is $$s = 2 + 15(1) - \frac{1}{2}(9.8)(1)^2 = 6.1 , \text{m}$$, and the velocity of the ball is $$v = 15 - 9.8(1) = 5.2 , \text{m/s}$$.
Pair 3: At $$t = 1.5 , \text{s}$$, the height of the ball is $$s = 2 + 15(1.5) - \frac{1}{2}(9.8)(1.5)^2 = 3.65 , \text{m}$$, and the velocity of the ball is $$v = 15 - 9.8(1.5) = 0.5 , \text{m/s}$$.

These pairs represent possible height and velocity pairs of the ball at some point in time.

Conclusion

In conclusion, when a ball is thrown vertically upward from a height of 2 m above the ground with an initial velocity of $$+15 , \text{m/s}$$, the possible height and velocity pairs of the ball at some point in time can be found using the equations of motion. The pairs represent the height and velocity of the ball at specific points in time and can be used to analyze the motion of the ball.

Discussion

The discussion category for this article is physics, as it deals with the motion of an object under the influence of gravity. The article provides a detailed analysis of the motion of the ball and presents possible height and velocity pairs of the ball at some point in time.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage Learning.

Note: The references provided are for general information purposes only and are not specific to the article.

Introduction

In our previous article, we explored the motion of a ball thrown vertically upward from a height of 2 m above the ground with an initial velocity of $$+15 , \text{m/s}$$. We analyzed the possible height and velocity pairs of the ball at some point in time using the equations of motion. In this article, we will answer some frequently asked questions related to the motion of the ball.

Q&A

Q1: What is the maximum height reached by the ball?

A1: The maximum height reached by the ball can be found using the equation $$s = s_0 + \frac{v_0^2}{2g}$$. Plugging in the values, we get $$s = 2 + \frac{(15)^2}{2(9.8)} = 7.69 , \text{m}$$.

Q2: What is the velocity of the ball at the maximum height?

A2: The velocity of the ball at the maximum height is zero, since the ball momentarily comes to rest at the maximum height.

Q3: How long does it take for the ball to reach the maximum height?

A3: The time it takes for the ball to reach the maximum height can be found using the equation $$t = \frac{v_0}{g}$$. Plugging in the values, we get $$t = \frac{15}{9.8} = 1.53 , \text{s}$$.

Q4: What is the acceleration of the ball at any point in time?

A4: The acceleration of the ball at any point in time is $$-9.8 , \text{m/s}^2$$, since the ball is under the influence of gravity.

Q5: What is the velocity of the ball when it hits the ground?

A5: The velocity of the ball when it hits the ground can be found using the equation $$v = v_0 + at$$. Plugging in the values, we get $$v = 15 - 9.8(1.53) = -9.45 , \text{m/s}$$.

Q6: How long does it take for the ball to hit the ground?

A6: The time it takes for the ball to hit the ground can be found using the equation $$t = \frac{2s}{v_0 + \sqrt{v_0^2 + 2gs}}$$. Plugging in the values, we get $$t = \frac{2(2)}{15 + \sqrt{(15)^2 + 2(9.8)(2)}} = 2.53 , \text{s}$$.

Conclusion

In conclusion, the Q&A article provides answers to some frequently asked questions related to the motion of a ball thrown vertically upward from a height of 2 m above the ground with an initial velocity of $$+15 , \text{m/s}$$. The answers provide a detailed analysis of the motion of the ball and can be used to understand the behavior of the ball at different points in time.

Discussion

The discussion category for this article is physics, as it deals with the motion of an object under the influence of gravity. The article provides a detailed analysis of the motion of the ball and answers some frequently asked questions related to the motion of the ball.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics (10th ed.). John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for Scientists and Engineers (10th ed.). Cengage Learning.

Note: The references provided are for general information purposes only and are not specific to the article.