Motion in One Dimension: Physics with Algebra Study Notes

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Motion in One Dimension

Introduction to Linear Motion

Motion in one dimension refers to the movement of objects along a straight line, either horizontally or vertically. This chapter focuses on describing, representing, and analyzing such motion using position, velocity, and acceleration.

Representing Position

Coordinate Axes and Position

To analyze motion, we use coordinate axes:

x-axis: Used for horizontal motion; positive direction is to the right.
y-axis: Used for vertical motion; positive direction is upward.

Position is specified relative to the origin (0 point) on the axis.

Position relative to origin on x and y axes

Motion Diagrams

Motion diagrams use dots to represent an object's position at successive times. These diagrams help visualize how an object moves over time.

Student's position at different times Motion diagram with equally spaced dots

Describing Motion with Graphs

Position-Versus-Time Graphs

Position-versus-time graphs (x vs t) plot an object's location over time. Key information can be extracted:

Position at time t: Read directly from the graph.
Velocity at time t: Determined by the slope of the graph at that point.
Direction of motion: Positive slope means motion to the right/up; negative slope means motion to the left/down.

Position vs time graph with slope changes Slope calculation on position vs time graph

Velocity-Versus-Time Graphs

Velocity-versus-time graphs (v vs t) show how an object's speed changes over time. The slope of this graph gives the object's acceleration.

Steeper slope: Faster speed.
Zero slope: Object at rest.
Negative slope: Object moving in the opposite direction.

Position and velocity graphs

Converting Between Graphs

The slope of the position-versus-time graph gives velocity, and the slope of the velocity-versus-time graph gives acceleration.

Position vs time graph with changing slopes Velocity vs time graph corresponding to position graph

Uniform Motion

Definition and Representation

Uniform motion, or constant-velocity motion, occurs when equal displacements happen during equal time intervals. The position-versus-time graph for uniform motion is a straight line.

Displacement: Change in position over time.
Velocity: Constant for uniform motion.

Uniform motion diagram and graph

Equations of Uniform Motion

The velocity of an object in uniform motion tells us how much its position changes each second:

Displacement formula:

Area under velocity graph gives displacement

Instantaneous Velocity

Definition and Calculation

Instantaneous velocity is the speed and direction of an object at a specific instant. If velocity changes, the position graph is curved, and the slope at a point (tangent) gives the instantaneous velocity.

Instantaneous velocity:

Curved position graph and tangent slope

Acceleration

Definition and Units

Acceleration describes how velocity changes over time. It is the slope of the velocity-versus-time graph.

Acceleration formula:
SI unit: meters per second squared (m/s2)

Acceleration as slope of velocity graph Example of animal acceleration Calculation of runner's speed

Sign of Acceleration

The sign of acceleration depends on the direction of motion and whether the object is speeding up or slowing down.

Positive acceleration: Speeding up in positive direction or slowing down in negative direction.
Negative acceleration: Speeding up in negative direction or slowing down in positive direction.

Acceleration sign and direction

Motion with Constant Acceleration

Kinematic Equations

For motion with constant acceleration, several equations relate position, velocity, acceleration, and time:

Velocity vs time graph for constant acceleration Area under velocity graph for displacement Displacement and velocity relationship Velocity changes steadily Position changes as square of time interval Change in velocity in terms of distance

Example: Braking to a Stop

When a car brakes to a stop, its velocity decreases steadily. The kinematic equations can be used to find acceleration and displacement.

Initial speed:
Final speed:
Time to stop:

Car braking motion diagram and velocity graph Given values for braking example Kinematic equations for braking Calculation of displacement while braking

Example: Minimum Runway Length for Takeoff

A plane accelerates to reach takeoff speed. The minimum runway length and time required can be calculated using kinematic equations.

Initial velocity:
Acceleration:
Final velocity:

Plane acceleration diagram Calculation of time to reach takeoff speed Calculation of minimum runway length

Free Fall

Definition and Properties

Free fall occurs when an object moves under the influence of gravity alone. All objects in free fall have the same acceleration, regardless of mass, if air resistance is negligible.

Free-fall acceleration: (downward)
g is always positive in calculations; direction is indicated by sign in equations.

Free fall motion diagram

Example: Analyzing a Rock's Fall

A rock dropped from rest falls 100 m. The time to fall and velocity upon impact can be found using kinematic equations for constant acceleration.

Initial position:
Initial velocity:
Acceleration:

Rock falling diagram and known values

Summary Table: Kinematic Equations for One-Dimensional Motion

Equation	Variables	Use
	Position, initial velocity, acceleration, time	Find final position
	Initial/final velocity, acceleration, time	Find final velocity
	Initial/final velocity, acceleration, displacement	Find velocity or displacement

Example: Use these equations to solve problems involving cars braking, planes taking off, or objects in free fall.