Position-Time and Velocity-Time Graphs: Space, Time, and Motion

Study Guide - Smart Notes

Tailored notes based on your materials, expanded with key definitions, examples, and context.

Space, Time, and Motion

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. The fundamental concepts include position, displacement, velocity, and acceleration. Graphical representations such as position-time and velocity-time graphs are essential tools for analyzing motion.

Position-Time and Velocity-Time Graphs

Position-Time Graphs

A position-time graph plots an object's position (x) as a function of time (t). The slope of the graph at any point represents the object's velocity.

Displacement (): The change in position of an object, calculated as .
Average Velocity (): The total displacement divided by the total time interval: .
Instantaneous Velocity (): The velocity at a specific instant, given by the slope of the tangent to the curve at that point: .
Interpretation: A straight line indicates constant velocity; a curved line indicates changing velocity (acceleration).

Example: If an object's position changes from 1.0 m to -3.0 m over 10 seconds, its average velocity is m/s.

Velocity-Time Graphs

A velocity-time graph plots velocity (v) as a function of time (t). The slope of the graph represents acceleration, and the area under the curve represents displacement.

Average Acceleration (): The change in velocity divided by the time interval: .
Instantaneous Acceleration (): The acceleration at a specific instant, given by the slope of the tangent to the curve at that point: .
Displacement (): The area under the velocity-time graph between two time points.

Example: If velocity changes from 2 m/s to -1 m/s over 3 seconds, average acceleration is m/s2.

Calculating Displacement from Velocity-Time Graphs

Displacement can be found by calculating the area under the velocity-time graph. For constant velocity, this is a rectangle; for changing velocity, it may be a combination of rectangles and triangles.

Rectangle Area:
Triangle Area:
Total Displacement: Sum the areas for each segment.

Example: For a velocity-time graph with a rectangle (2 m/s for 5 s) and a triangle (change from 2 m/s to 4 m/s over 2 s):

Worked Examples from Notes

Finding Position at a Given Time: Read the value from the position-time graph at the specified time.
Average Velocity Calculation:
Instantaneous Velocity: Find the slope of the tangent at the required time.
Average Acceleration Calculation:
Instantaneous Acceleration: Find the slope of the tangent to the velocity-time graph at the required time.
Displacement Over an Interval: Sum the areas under the velocity-time graph for the interval.

Summary Table: Key Quantities in Kinematics

Quantity	Definition	Graphical Representation
Displacement ()	Change in position	Vertical difference on position-time graph
Average Velocity ()	Displacement per unit time	Slope of secant line on position-time graph
Instantaneous Velocity ()	Velocity at a specific instant	Slope of tangent line on position-time graph
Average Acceleration ()	Change in velocity per unit time	Slope of secant line on velocity-time graph
Instantaneous Acceleration ()	Acceleration at a specific instant	Slope of tangent line on velocity-time graph
Displacement from Velocity-Time Graph	Area under the curve	Sum of areas (rectangles, triangles)

Additional info:

Position-time and velocity-time graphs are foundational for understanding motion in one dimension.
Instantaneous quantities are found using the tangent (derivative) at a point, while average quantities use secant (difference) over intervals.
Displacement can be positive or negative, depending on direction.
Velocity and acceleration can also be positive or negative, indicating direction of motion and change.