One-Dimensional Motion & Kinematics

Fundamental Motion Quantities

Understanding motion in one dimension requires precise definitions of physical quantities that describe how objects move. These quantities are foundational for analyzing and solving kinematics problems.

• Position (): The location of an object relative to a reference point or origin. Unit: meters ().

• Distance (): The total path length traveled by an object. Distance is a scalar and always non-negative. Unit: meters ().

• Displacement (): The overall change in position of an object. Includes both magnitude and direction. Unit: meters ().

• Speed ( or ): The rate at which an object covers distance. Speed is scalar. Unit: meters per second ().

• Velocity (): The rate of change of position. Velocity is a vector (includes direction). Unit: meters per second ().

• Acceleration (): The rate at which velocity changes over time. Acceleration is a vector. Unit: meters per second squared ().

Scalar vs. Vector Quantities

Physical quantities are classified as either scalars or vectors, depending on whether they include direction.

• Scalar Quantities: Described by magnitude (numerical value and units) only. No direction involved. Examples: Distance, Speed, Mass, Time, Temperature.

• Vector Quantities: Described by both magnitude and direction. Direction can be indicated by cardinal points or sign conventions ( or ). Examples: Position, Displacement, Velocity, Acceleration, Force.

Frame of Reference

Motion is always described relative to a chosen frame of reference, which consists of a coordinate system and an origin point.

• Definition: A frame of reference is used to specify and measure positions and motion of objects.

• Relative Motion: An object's motion can appear different depending on the observer's frame. For example, a passenger sitting in a moving train is stationary relative to the train but moving relative to the ground.

• Coordinate Sign Conventions:

  • Horizontal 1D motion: Right/East = Positive (); Left/West = Negative ().

  • Vertical 1D motion: Upward = Positive (); Downward = Negative ().

Kinematic Equations & Motion Problem Solving

For objects moving with constant acceleration, the following kinematic equations (Uniformly Accelerated Motion, UAM) are used to relate displacement, velocity, acceleration, and time.

• For constant velocity ():

G-U-E-S-S Problem-Solving Framework:

1. G — Given: List all known variables with units and directional signs ().

2. U — Unknown: Identify the target variable to calculate.

3. E — Equation: Select the appropriate kinematic equation.

4. S — Substitute: Plug known values into the formula.

5. S — Solve & State: Calculate, state units and direction, and check for reasonableness.

Free Fall: Special Case of Vertical Motion

Free fall describes motion under the influence of gravity alone, with negligible air resistance.

• Acceleration due to gravity: downward.

• If upward is positive: . If downward is positive: .

• All objects in free fall accelerate at the same rate, regardless of mass.

• At the peak of an upward throw, instantaneous velocity is zero (), but acceleration remains .

Example Problems

• Problem 1 (Horizontal UAM): A car starting from rest accelerates uniformly at for . Find final velocity and displacement.

  • Given: , ,

  • Final Velocity: forward.

  • Displacement: .

• Problem 2 (Free Fall): A stone is dropped from rest from a bridge and hits the water after . Find impact velocity and bridge height (downward positive).

  • Given: , ,

  • Impact Velocity: downward.

  • Bridge Height: .

Graphing Motion: Position–Time and Velocity–Time Graphs

Graphical analysis is a powerful tool for visualizing and interpreting motion. Two common graphs are position–time (–) and velocity–time (–$t$) graphs.

Position–Time (–) Graphs

• Axes: Vertical axis = Position (), Horizontal axis = Time ().

• Slope: .

• Positive slope: Moving in positive direction.

• Negative slope: Moving in negative direction.

• Zero slope: Object is stationary ().

• Steeper slope: Greater speed.

• Straight line: Constant velocity ().

• Curved line: Changing velocity (non-zero acceleration).

• Curved upward (concave up): Positive acceleration.

• Curved downward (concave down): Negative acceleration.

Velocity–Time (–) Graphs

• Axes: Vertical axis = Velocity (), Horizontal axis = Time ().

• Slope: .

• Horizontal line above zero: Constant positive velocity ().

• Upward sloping line: Positive acceleration.

• Downward sloping line: Negative acceleration.

• Crossing the time axis (): Object changes direction.

• Area Under Curve: Represents displacement ().

• Area above time axis (): Positive displacement.

• Area below time axis (): Negative displacement.

• Rectangular region: .

• Triangular region: .

Graph Interpretation Summary Table