Introduction

Relevance of the Topic

Kinematics: Vertical Motion is highly relevant as it is the foundation for understanding more complex concepts in physics and other disciplines. The vertical nature of movements occurs in various situations in our daily lives, from basketball throws to the movement of planets in the solar system.

Contextualization

This topic is an integral part of the Physics discipline, specifically in the domain of kinematics. Vertical motion is a basic and essential concept that connects and extends to more advanced topics such as oblique launches, circular motion, and even dynamics (force and acceleration). It serves as a starting point for understanding more complex concepts and is therefore crucial for the development of problem-solving skills in physics.

This topic will be especially useful in the study of future topics such as Newton's laws of motion and principles of conservation of energy, as they are intrinsically linked to vertical motion concepts.

Theoretical Development

Components

• Gravity: The force that attracts an object towards the center of the Earth. In vertical motion, gravity is the predominant force acting on the object, resulting in accelerated motion.

• Vertical Motion Equation (without air resistance): The equation that describes the position of an object in vertical motion as a function of time. The equation is usually given by: y(t) = y0 + v0t - (1/2)gt^2, where y(t) is the object's position at time t, y0 is the initial position, v0 is the initial velocity, and g is the acceleration due to gravity (-9.8m/s^2 on Earth).

• Vertical Launch Up and Down: The direction of the initial velocity influences the motion's direction. When the initial velocity is upwards, the object decelerates until reaching the maximum height and then starts to fall. When the initial velocity is downwards, the object accelerates as it falls.

• Time of Ascent, Descent, and Total: The total time of vertical motion is the time it takes for the object to go from the lowest point to the highest point and then back to the lowest point. The time of ascent is the time it takes for the object to go from the lowest point to the highest point, and the time of descent is the time it takes for the object to return from the highest point to the lowest point. The time of ascent and descent are equal.

• Maximum and Minimum Velocity: The maximum velocity is reached during the fall and the minimum velocity during the ascent. The maximum velocity is always greater than the initial velocity, and the minimum velocity is always less than the initial velocity (unless the initial velocity is zero, where the minimum velocity will also be zero).

Key Terms

• Vertical Motion: It is any motion that occurs along a straight line in the vertical direction, i.e., upwards or downwards.

• Acceleration due to Gravity: This is the acceleration that an object in free fall (or near the Earth's surface) experiences due to gravity. On Earth, the acceleration due to gravity is commonly represented as 9.8 m/s².

• Initial Velocity: It is the value of the velocity of an object at the beginning of the motion.

• Initial Position: It is the position of an object at the beginning of the motion.

• Equation of Motion for Uniformly Accelerated Motion: It is the equation that describes the position of an object as a function of time when this object is under the action of accelerated motion. The equation is given by s = s0 + vt + (1/2)at^2, where s is the position at time t, s0 is the initial position, v is the velocity, a is the acceleration, and t is the time.

Examples and Cases

1. Vertical Launch Up without Initial Velocity: An object is vertically launched upwards from the ground. After some time, it stops in the air and starts to fall. We can calculate the time it takes to reach the maximum height using the equation v = v0 - gt, where v is the final velocity (zero at the stopping point), v0 is the initial velocity (to be found), and g is the acceleration due to gravity. Then, we can find the maximum height using the equation h = h0 + v0t - (1/2)gt^2, where h is the maximum height (unknown), h0 is the initial height (zero), and t is the time (previously calculated).

2. Vertical Launch Down with Initial Upward Velocity: Now, an object is vertically launched downwards from the maximum height reached in the previous example, but with an initial upward velocity. We can use the same equation to find the time it takes to reach the ground. However, the initial velocity is now positive, which means that the acceleration due to gravity will decrease its magnitude. This is an important example to understand the difference between initial and minimum velocity - the initial velocity is positive, but the minimum velocity (at the highest point) is zero.

3. Vertical Launch Down with Initial Downward Velocity: In this case, the object is dropped from the maximum height with an initial downward velocity. The initial velocity is still positive, but the acceleration due to gravity will increase its magnitude, resulting in a faster movement towards the ground. This example illustrates that gravity always acts in the opposite direction to the velocity, resulting in deceleration or acceleration, depending on the direction of the initial velocity.

Detailed Summary

Key Points

• Influence of Gravity: Gravity is the primary force that drives vertical motion. It enhances or counteracts the initial forces, determining the direction and speed of the movement.

• Vertical Motion Equation: The equation represents the object's trajectory in vertical motion. It considers the initial position, initial velocity, time, and gravity to determine the object's position at any given moment.

• Launches Up and Down: The direction of the initial velocity directly influences the movement's direction. When the object is launched upwards, gravity acts as a force against the velocity, decelerating the object. In a downward launch, gravity and velocity act in the same direction, resulting in accelerated movement.

• Time of Ascent, Descent, and Total: The total time of vertical motion is the sum of the ascent and descent times. However, when calculating the ascent and descent times separately, they will be equal.

• Maximum and Minimum Velocity: The initial velocity, when upwards, becomes the maximum velocity during the fall. However, at the point of reaching the highest point, the velocity becomes minimum, and after that point, the object starts to accelerate again towards the ground.

Conclusions

• Vertical Motion: The study of vertical motion kinematics provides the fundamental concepts to understand other topics in physics and their application in explaining everyday phenomena.

• Context Importance: The direction and magnitude of the initial velocity, along with the action of gravity, are crucial to fully understand vertical motion.

• Universal Physical Laws: The laws of physics, such as Newton's law of gravity, are applicable to all bodies anywhere in the universe, and it is through the study of vertical motion that we begin to explore these laws.

Suggested Exercises

1. Exercise 1: An object is vertically launched upwards with an initial velocity of 20 m/s. Determine the time it takes for the object to reach the maximum height and the maximum height reached.

2. Exercise 2: An object is dropped from a platform at a height of 100 m from the ground. Determine the velocity at which the object hits the ground.

3. Exercise 3: An object is vertically launched downwards from rest from a height of 50 m. Determine the time it takes for the object to reach the ground and the velocity at which it hits the ground.