Displacement, Velocity, and Acceleration

Introduction

Key Concepts

Displacement

Displacement is a vector quantity that denotes the change in position of an object. It is defined as the straight-line distance from the initial position to the final position, along with the direction of movement. Unlike distance, which is scalar and only accounts for the magnitude of movement, displacement provides both magnitude and direction, making it essential for vector analysis in physics.

The mathematical representation of displacement ($\vec{d}$) can be expressed as: $$ \vec{d} = \vec{d_f} - \vec{d_i} $$ where $\vec{d_f}$ is the final position vector and $\vec{d_i}$ is the initial position vector.

**Example:** If a car moves from point A to point B, 100 meters east, its displacement is 100 meters east, regardless of the path taken.

Velocity

Velocity is a vector quantity that describes the rate of change of displacement with respect to time. It indicates both how fast an object is moving and the direction of its movement. Velocity is different from speed, which is scalar and only measures how fast an object is moving without considering direction.

The average velocity ($\vec{v}$) is calculated using the formula: $$ \vec{v} = \frac{\vec{d}}{t} $$ where $\vec{d}$ is displacement and $t$ is the time taken.

Instantaneous velocity is the velocity of an object at a specific moment in time and can be determined using calculus: $$ \vec{v} = \frac{d\vec{d}}{dt} $$

**Example:** If a cyclist travels 50 meters north in 10 seconds, the average velocity is 5 meters per second north.

Acceleration

Acceleration is a vector quantity that measures the rate of change of velocity with respect to time. It indicates how an object's velocity is increasing or decreasing. Positive acceleration denotes an increase in velocity, while negative acceleration (deceleration) indicates a decrease.

The average acceleration ($\vec{a}$) is given by: $$ \vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v_f} - \vec{v_i}}{t} $$ where $\Delta \vec{v}$ is the change in velocity and $\Delta t$ is the change in time.

Instantaneous acceleration can be found using the derivative of velocity with respect to time: $$ \vec{a} = \frac{d\vec{v}}{dt} $$

**Example:** If a car's velocity increases from 20 m/s to 30 m/s in 5 seconds, the acceleration is 2 m/s².

Equations of Motion

In kinematics, the equations of motion relate displacement, initial velocity, final velocity, acceleration, and time. For uniformly accelerated motion, the following equations are fundamental:

• $$ \vec{v} = \vec{u} + \vec{a}t $$ where $\vec{v}$ is the final velocity, $\vec{u}$ is the initial velocity, and $t$ is time.
• $$ \vec{d} = \vec{u}t + \frac{1}{2}\vec{a}t^2 $$
• $$ \vec{v}^2 = \vec{u}^2 + 2\vec{a}\vec{d} $$

These equations are instrumental in solving problems involving linear motion with constant acceleration.

Graphical Analysis

Displacement, velocity, and acceleration can be analyzed graphically, providing visual insights into motion. Common graphs include:

• Displacement-Time Graph: The slope represents velocity.
• Velocity-Time Graph: The slope represents acceleration, while the area under the curve represents displacement.
• Acceleration-Time Graph: The area under the curve represents the change in velocity.

These graphical representations aid in interpreting motion scenarios and extracting quantitative information.

Relative Motion

Relative motion examines the movement of an object as observed from different frames of reference. Displacement, velocity, and acceleration can vary depending on the observer's perspective. Understanding relative motion is essential for analyzing scenarios where multiple objects are in motion relative to each other.

For two frames of reference moving with velocities $\vec{v_1}$ and $\vec{v_2}$, the relative velocity ($\vec{v_{rel}}$) is: $$ \vec{v_{rel}} = \vec{v_2} - \vec{v_1} $$

**Example:** A person walking on a moving train perceives their movement differently compared to an observer on the platform.

Applications in Physics SL

Displacement, velocity, and acceleration are applied in various physics problems, including projectile motion, circular motion, and free-fall scenarios. Mastery of these concepts enables students to solve complex problems and understand the underlying principles governing motion.

In the IB Physics SL curriculum, these concepts are integrated into experiments and theoretical investigations, fostering a deep comprehension of kinematic principles.

Comparison Table

Summary and Key Takeaways