Motion in One Dimension §

Introduction §

Motion in one dimension refers to the movement of objects along a straight line. This fundamental concept in physics is crucial for understanding more complex motions in two and three dimensions.

Key Concepts §

Displacement §

• Definition: Displacement is the change in the position of an object. It is a vector quantity, which means it has both magnitude and direction.
• Formula: $\Delta x=x_f-x_i$, where $x_f$ is the final position and $x_i$ is the initial position.
• Units: Measured in meters (m) in the International System of Units (SI).

Velocity §

• Definition: Velocity is the rate of change of displacement with respect to time. It is a vector quantity.
• Formula: $v=\frac{\Delta x}{\Delta t}$, where $\Delta x$ is displacement and $\Delta t$ is the time interval.
• Average Velocity: $\bar{v}=\frac{x_f-x_i}{t_f-t_i}$.
• Units: Measured in meters per second (m/s).

Speed §

• Definition: Speed is the magnitude of velocity. It is a scalar quantity and represents how fast an object is moving.
• Formula: Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$.
• Units: Measured in meters per second (m/s).

Acceleration §

• Definition: Acceleration is the rate of change of velocity with respect to time. It is a vector quantity.
• Formula: $a=\frac{\Delta v}{\Delta t}$, where $\Delta v$ is the change in velocity and $\Delta t$ is the time interval.
• Units: Measured in meters per second squared (m/s²).

Historical Context §

The study of motion in one dimension has been crucial in the development of classical mechanics. Galileo Galilei and Sir Isaac Newton were pivotal figures in establishing the principles governing motion.

Examples §

Example 1: Free Fall §

A classic example of motion in one dimension is free fall, where an object is subject to gravity alone, typically near the Earth’s surface. The acceleration due to gravity is approximately $9.8{\text{m/s}}^2$.

Example 2: Cars on a Straight Road §

Consider cars moving along a straight road. Their motion can be analyzed in terms of displacement, velocity, and acceleration along this one-dimensional path.

Test Questions §

1. STARTI [Basic] Question: What is the formula for calculating average velocity? Back: Average velocity is calculated using the formula $\bar{v}=\frac{x_f-x_i}{t_f-t_i}$, where $x_f$ and $x_i$ are the final and initial positions, respectively, and $t_f$ and $t_i$ are the final and initial times. ENDI
2. STARTI [Basic] Question: If a car accelerates from rest to $60\text{m/s}$ in $5\text{seconds}$, what is its acceleration? Back: The acceleration $a$ is given by $a=\frac{\Delta v}{\Delta t}=\frac{60\text{m/s}}{5\text{s}}=12{\text{m/s}}^2$. ENDI
3. STARTI [Basic] Question: Describe the difference between speed and velocity. Back: Speed is a scalar quantity that measures how fast an object is moving. Velocity, on the other hand, is a vector quantity that includes both the speed and direction of the object’s motion. ENDI

For further exploration and in-depth understanding, refer to the links embedded in the note and review the test questions regularly.