It would be better for the true physics if there were no mathematicians on earth.
-Daniel Bernoulli
Definition of Motion: What is It and How Is It Described?
Motion can be simply defined as the change in the position of an object over time. If an object changes its position during a given period, it is in motion. However, to define motion, a reference point is necessary because motion is the change of something relative to something else. For a passenger sitting inside a moving train, the train may be considered stationary; however, if we consider the train in relation to the Earth, the passenger is also in motion relative to the Earth.
Kinematics describes motion using three fundamental quantities: Position, Velocity, and Acceleration. These quantities are used to detail and precisely describe the motion of an object.
Kinematic Quantities: Tools for Describing Motion
1. Position: Where Is It?
Position describes the location of an object relative to a reference point. It is usually expressed using a coordinate system. For example, in a plane, the position of an object can be defined with a two-dimensional coordinate system ((x, y)); in three-dimensional space, it can be identified with a point ((x, y, z)).
Example:
To describe the location of a person in a city, latitude and longitude coordinates can be used. If the person is at 41° N latitude and 29° E longitude, their position is defined relative to a point on Earth. This is a practical example of how position is used in kinematics.
2. Displacement: How Far?
Displacement is the linear distance and direction from an object’s initial position to its final position. Displacement is a vector quantity, meaning it has both magnitude and direction. Unlike the scalar quantity distance, displacement only defines the shortest path between the start and end points of the motion.
Example:
If a student leaves home and walks 300 meters north on a straight road and then 400 meters east, the total distance covered is 700 meters. However, the displacement can be calculated using the Pythagorean theorem:
$$
\Delta x = \sqrt{300^2 + 400^2} = 500 \, \text{meters}
$$
The displacement, in this case, is 500 meters in the northeast direction.
3. Velocity: How Fast?
Velocity is a vector quantity that describes how much an object’s displacement changes over a unit of time, incorporating both magnitude and direction. Unlike the scalar quantity speed, velocity is defined as the ratio of the displacement vector to time. Mathematically, velocity is defined as:
$$
\mathbf{v} = \frac{\Delta \mathbf{x}}{\Delta t}
$$
where (\mathbf{v}) is velocity, (\Delta \mathbf{x}) is displacement, and (\Delta t) is the elapsed time.
Example:
If a runner completes a 100-meter track in 10 seconds, the average velocity is:
$$
\mathbf{v} = \frac{100 \, \text{m}}{10 \, \text{s}} = 10 \, \text{m/s} \, \text{forward}
$$
This means the runner’s displacement is 10 meters per second.
4. Acceleration: How Does Velocity Change?
Acceleration is a vector quantity that describes how much an object’s velocity changes over a unit of time. If an object’s velocity changes over time, it is said to be accelerating. Acceleration is calculated as the change in velocity over time:
$$
\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t}
$$
where (\mathbf{a}) is acceleration, (\Delta \mathbf{v}) is the change in velocity, and (\Delta t) is the elapsed time.
Example:
If a car accelerates from rest (0 m/s) to 24 m/s in 8 seconds, the average acceleration is:
$$
\mathbf{a} = \frac{24 \, \text{m/s} – 0 \, \text{m/s}}{8 \, \text{s}} = 3 \, \text{m/s}^2
$$
This means the car’s speed increases by 3 m/s every second.
Types of Motion: Linear and Curvilinear Motion
Kinematics examines two basic types of motion: linear motion and curvilinear motion. Each type is defined by how the object moves and the path it follows.
1. Linear Motion:
Linear motion is the movement of an object along a straight line. This type of motion is analyzed in only one dimension, typically along the x-axis. When linear motion occurs with constant acceleration, the equations of motion are:
$$
v = v_0 + at
$$
$$
x = x_0 + v_0 t + \frac{1}{2} a t^2
$$
$$
v^2 = v_0^2 + 2a(x – x_0)
$$
where:
- \(v\): Final velocity
- \(v_0\): Initial velocity
- \(a\): Acceleration
- \(t\): Time elapsed
- \(x\) and \(x_0\): Final and initial positions
Example:
If a car starts moving with an initial velocity of 5 m/s and accelerates at a constant rate of 3 m/s² for 4 seconds, the final velocity will be:
$$
v = v_0 + at = 5 \, \text{m/s} + (3 \, \text{m/s}^2)(4 \, \text{s}) $$
$$= 17 \, \text{m/s}
$$
This equation shows how velocity changes over time.
2. Curvilinear Motion:
Curvilinear motion is the movement of an object along a curved path. It is typically analyzed in two or three-dimensional space, often expressed using vectors.
Example:
When a ball is thrown into the air at a certain angle and velocity, it follows a parabolic trajectory. In this case, the horizontal motion along the x-axis is at constant velocity (without acceleration), while the vertical motion along the y-axis changes under the influence of gravitational acceleration. The ball’s trajectory can be calculated by combining the motion equations for both axes:
\[
x = v_0 \cos(\theta) \cdot t, \quad y = v_0 \sin(\theta) \cdot t – \frac{1}{2} g t^2
\]
where:
- \(v_0\): Initial velocity
- \(\theta\): Launch angle
- \(g\): Gravitational acceleration (9.8 m/s²)
- \(t\): Time
To better understand this topic, it is helpful to practice problem-solving and explore real-world examples. Additionally, using various simulation tools and software to visualize different types of motion can make the concepts easier to grasp. Experiments and observations are also powerful methods to comprehend the fundamental principles of motion. Remember, kinematics is not just about formulas; it is a way to understand how we perceive and interact with the world.
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