Motion Along a Straight Line

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In this chapter, we focus on straight-line (one-dimensional) motion and introduce key physical quantities such as position, displacement, velocity, and acceleration.

• Position: The location of an object in a coordinate system.

• Displacement: The change in position of an object; a vector quantity.

• Velocity: The rate of change of displacement; can be average or instantaneous.

• Acceleration: The rate of change of velocity; can be average or instantaneous.

Coordinate Systems

A coordinate system is essential for describing the location and motion of objects. It consists of:

• A fixed reference point called the origin

• A set of axes (e.g., x, y, z in Cartesian coordinates)

• A definition of the coordinate variables

Distance and Displacement

Distance and displacement are fundamental concepts in describing motion:

• Distance: The total length of travel; always positive and measured by devices like an odometer.

• Displacement: The change in position, defined as ; can be positive, negative, or zero, and is a vector.

Analytic Representation of Motion

Instantaneous position, displacement, velocity, and acceleration can be expressed using coordinates and unit vectors:

• Position:

• Displacement:

Velocity and Acceleration

Velocity and acceleration are defined as derivatives:

• Velocity:

• Acceleration:

Linear (One-Dimensional) Motion

In one-dimensional motion, displacement is relative to the origin and can be expressed as (horizontal) or (vertical). The x-direction is often used as a prototype for analysis.

Average Speed and Velocity

Speed and velocity are distinct in physics:

• Average speed: (always positive)

• Average velocity: (can be positive, negative, or zero)

• SI units: m/s

Instantaneous Velocity

The velocity at a specific instant is found by taking the limit as the time interval approaches zero:

• Graphically, it is the slope of the tangent line to a position vs. time graph.

Average and Instantaneous Acceleration

Acceleration describes how velocity changes over time:

• Average acceleration:

• Instantaneous acceleration:

• Graphically, acceleration is the slope of a velocity vs. time curve.

• Unit: m/s2

Motion with Constant Acceleration

When acceleration is constant, the following equations describe the motion:

Deceleration

Deceleration refers to decreasing speed and occurs when velocity and acceleration have opposite signs. It is not the same as negative acceleration.

Freely Falling Objects

Objects in free fall experience constant acceleration due to gravity:

• downward

• Free fall is independent of mass (neglecting air resistance)

• Equations for free fall mirror those for constant acceleration, with

Graphical Representation of Motion

Motion can be represented in several ways:

• Position vs. time plots: Slope gives velocity

• Velocity vs. time plots: Slope gives acceleration

• Acceleration vs. time plots: Area under curve gives change in velocity

Summary Table: Equations for 1-D Motion with Constant Acceleration

Problem Solving Strategy

• List given quantities

• Make a sketch and draw coordinate axes

• Identify the positive direction

• Determine what is to be solved

• Be consistent with units

• Check the reasonableness of your answer

Examples

• Average speed at the equator:

• Average velocity at the equator: (since displacement after one rotation is zero)

• Car motion: If a car moves from +4 m to -1 m in 2 s, with initial velocity -4 m/s and final velocity -1 m/s:

  • Displacement: m

  • Average velocity: m/s

  • Average acceleration: m/s2

Important Rules

• Distinguish between vectors (velocity, displacement, acceleration) and scalars (speed, distance).

• Carefully differentiate between average and instantaneous quantities.

• Use correct sign conventions and units.

Additional info:

• Graphs and diagrams are essential for visualizing motion and interpreting physical quantities.

• All equations assume motion along a straight line (one dimension).