Kinematics Variables: Definitions, Applications, and Practice

Kinematics Variables

Introduction

Kinematics is a branch of mechanics that describes the motion of objects without considering the causes of motion. The study of kinematics involves several key variables, including position, displacement, velocity, speed, and acceleration. Understanding these variables is essential for analyzing and predicting the motion of objects in one dimension.

Kinematics versus Dynamics

Definitions and Distinctions

• Kinematics: Concerns the description of motion (position, velocity, acceleration) without reference to the forces causing the motion.

• Dynamics: Explains why motion occurs by considering the forces and interactions responsible for changes in motion.

• Mechanics: The combination of kinematics and dynamics, providing a complete framework for understanding motion and its causes.

Kinematics Variable Definitions

Position

Position describes an object's location with respect to a reference point, often called the origin (zero point). It is typically measured along a horizontal (x) or vertical (y) axis.

• Choose a convenient origin and define positive direction.

• Units are usually meters (m).

• Positive position: right of or above the origin.

• Negative position: left of or below the origin.

Displacement

Displacement is the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction.

• SI unit: meter (m).

• Formula: $\Delta x = x - x_0$

• Direction is indicated by the sign (positive or negative).

Practice Table: Displacement Calculation

Initial position, $x_0$	Final position, $x$
+1.7 m	+6.1 m
+5.6 m	+1.8 m
-3.2 m	+7.2 m

Displacement versus Distance Travelled

Distance travelled is the total length of the path taken, while displacement is the straight-line change in position from start to finish.

• Distance: always positive, scalar quantity.

• Displacement: can be positive or negative, vector quantity.

• Example: If an object moves 1 m, then 3 m, then 5 m, total distance = 9 m, but displacement may be different depending on direction.

• Formula for displacement: $\Delta x = x_3 - x_0$

Average Velocity and Average Speed

Average speed is a scalar equal to the distance travelled divided by the elapsed time. Average velocity is a vector equal to the displacement divided by the elapsed time.

• Average speed: $\text{average speed} = \frac{\text{distance travelled}}{\text{elapsed time}}$

• Average velocity: $\text{average velocity} = \frac{\text{displacement}}{\text{elapsed time}}$

• Formula: $\vec{v}_{\text{ave}} = \frac{\vec{x} - \vec{x}_0}{t - t_0} = \frac{\Delta x}{\Delta t}$

(Instantaneous) Velocity

Instantaneous velocity ($\vec{v}$) describes how fast and in what direction an object is moving at a specific instant.

• Speed is the magnitude of velocity.

• SI unit: meter per second (m/s).

• Formula: $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

• Direction is indicated by the sign.

Example Calculation

• Average speed: $\frac{9\,\text{m}}{3\,\text{s}} = 3\,\text{m/s}$

• Average velocity: $\frac{-1\,\text{m}}{3\,\text{s}} = -0.33\,\text{m/s}$

Practice Problem

A ball rolls in the positive direction of an x-axis for 25 seconds at a constant speed of 1.5 m/s. Then the ball rolls in the negative direction for 12 seconds at a constant speed of 2.0 m/s. What is the average velocity of the ball?

• A. +1.7 m/s

• B. -1.7 m/s

• C. +1.2 m/s

• D. +0.36 m/s

Average Acceleration and (Instantaneous) Acceleration

Average acceleration is a vector equal to the change in velocity divided by the time required for the change to occur. Instantaneous acceleration describes how fast velocity changes at a specific instant.

• Average acceleration: $\text{average acceleration} = \frac{\text{change in velocity}}{\text{elapsed time}}$

• Formula: $\vec{a}_{\text{ave}} = \frac{\vec{v} - \vec{v}_0}{t - t_0} = \frac{\Delta v}{\Delta t}$

• Instantaneous acceleration: $\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}$

• SI unit: meter per second squared (m/s2).

Acceleration versus Deceleration

Acceleration and deceleration depend on the direction of velocity and acceleration vectors:

• If velocity and acceleration are in the same direction, the object accelerates.

• If velocity and acceleration are in opposite directions, the object decelerates.

Practice Problem

A wombat moves along the x-axis. What is the direction of the acceleration if it is moving:

• a) In the positive direction with increasing speed? Positive

• b) In the positive direction with decreasing speed? Negative

• c) In the negative direction with increasing speed? Negative

• d) In the negative direction with decreasing speed? Positive

Homework and Practice

Assignments

• Quiz Q04a – Kinematics Variables (due following Tuesday class)

• Kinematics Equations, Free-fall: read sections 2.5–7

Extra Practice Questions

• Displacement: 2.1 (7 m, 7 m, +7 m), 2.2 (5 m, 5 m, -5 m), 2.3 (12 m, 8 m, +8 m), 2.4 (8 m, 4 m, -4 m)

• Time, Velocity, and Speed: 2.11 (40.0 km/h, 34.3 km/h 25° S of E, 3.20 km/h), 2.14 (6.00 m/s, -1.71 m/s, 4.04 m/s, 3.29 m/s)

• Acceleration: 2.16 (4.29 m/s2), 2.17 (56.4 m/s2, 5.76g; -201 m/s2, 20.6g), 2.18 (1.43 s, -2.50 m/s2), 2.19 (108 m/s2, 11.1g)

Additional info: These notes provide foundational definitions and practice for kinematics variables, suitable for introductory college physics courses.