Foundations of Physics: Motion, Velocity, and Acceleration

Introduction to Motion in Physics

This study guide covers foundational concepts in kinematics, focusing on motion diagrams, average speed and velocity, acceleration, and the use of units and significant figures. These topics are essential for understanding how objects move and how to quantitatively describe their motion in physics.

Motion Diagrams

Definition and Purpose

A motion diagram is a visual representation that shows the position of an object at several equally spaced instants of time. It helps in analyzing and understanding the motion of objects by providing a sequence of positions.

• Rest: An object at rest is represented by a single position in the diagram (e.g., a stationary ball on the ground).

• Constant Speed: Equally spaced images indicate constant speed (e.g., a skateboarder rolling at a steady pace).

• Speeding Up: Increasing distance between images shows acceleration (e.g., a sprinter starting a race).

• Slowing Down: Decreasing distance between images shows deceleration (e.g., a car stopping for a red light).

• Complex Motion: Some diagrams show both speeding up and slowing down (e.g., a jump shot in basketball).

Example: A motion diagram of a rocket launch shows the rocket's position at successive time intervals, with numbers indicating the order of frames.

Average Speed and Velocity

Definitions

• Average Speed: The total distance traveled divided by the total time taken. Equation: $\text{average speed} = \frac{\text{distance traveled}}{\text{time interval spent traveling}} = \frac{d}{\Delta t}$

• Average Velocity: The displacement (change in position) divided by the time interval. Equation: $\vec{v}_{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t}$

Key Points:

• Speed is a scalar quantity (magnitude only), while velocity is a vector (magnitude and direction).

• If the direction of motion does not change, the magnitude of average velocity equals the average speed.

• If the direction changes (e.g., running around a track and returning to the start), the displacement is zero, so average velocity is zero, but average speed is nonzero.

Example: Running around a 38 m track in 10 s: average speed = 3.8 m/s, but average velocity = 0 m/s (since displacement is zero).

Acceleration

Linear Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity, so it can change in magnitude (speed) or direction.

• Average Acceleration: Change in velocity divided by the time interval. Equation: $\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t}$

• Acceleration can mean speeding up, slowing down, or changing direction.

• When velocity and acceleration vectors point in the same direction, the object speeds up. When they point in opposite directions, the object slows down.

Example: If a car increases its velocity from 10 m/s to 30 m/s in 10 s, the average acceleration is: $\vec{a}_{\text{avg}} = \frac{30\,\text{m/s} - 10\,\text{m/s}}{10\,\text{s}} = 2.0\,\text{m/s}^2$

Position vs. Time Graphs

Interpreting Graphs

Position vs. time graphs are useful for visualizing motion. The slope of the line on such a graph represents the velocity.

• Straight Line: Indicates constant velocity; the slope equals the velocity.

• Curved Line: Indicates changing velocity (acceleration).

Example: A student walking to school has a position vs. time graph with dots at equal time intervals, showing their changing position.

Units and Dimensional Analysis

Fundamental and Derived Units

• Fundamental Units:

  • Length: meter (m)

  • Time: second (s)

  • Mass: kilogram (kg)

• Derived Units:

  • Area: $\text{m}^2$

  • Volume: $\text{m}^3$

  • Velocity: $\text{m/s}$

  • Acceleration: $\text{m/s}^2$

Unit Systems:

• SI (mks): meters, kilograms, seconds

• cgs: centimeters, grams, seconds

• British: inches, feet, miles, pounds

Unit Conversion: Always use appropriate conversion factors (e.g., 1 inch = 2.54 cm, 1 mile = 1.61 km).

Dimensional Analysis

• Both sides of an equation must have the same units.

• Only quantities with the same dimensions can be added or subtracted.

• Checking units helps catch mistakes and verify answers.

Significant Figures

Rules and Importance

• Measurements always have some uncertainty; significant figures reflect this precision.

• When combining values, the result should not have more significant figures than the least precise measurement.

• Trailing zeros in whole numbers are not significant unless specified by a decimal point or scientific notation.

Example: 320 has two significant figures; 3.20 × 102 has three significant figures.

Summary Table: Key Kinematic Quantities

Quantity	Definition	SI Unit	Equation
Distance	Total length traveled	meter (m)	—
Displacement	Change in position (vector)	meter (m)	$\Delta \vec{x} = \vec{x}_f - \vec{x}_i$
Speed	Distance per unit time (scalar)	m/s	$\text{speed} = \frac{d}{\Delta t}$
Velocity	Displacement per unit time (vector)	m/s	$\vec{v} = \frac{\Delta \vec{x}}{\Delta t}$
Acceleration	Change in velocity per unit time (vector)	m/s2	$\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$

Additional info: Some context and examples were expanded for clarity and completeness, as is standard in academic study guides.