Kinematics

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. It focuses on how objects move, including their position, velocity, and acceleration, typically starting with one-dimensional (1D) motion before extending to two-dimensional (2D) motion.

• Kinematics: The study of the description of motion.

• We first analyze motion in 1D, then extend to 2D scenarios.

Reference Frames

Defining Reference Frames

A reference frame is a system for specifying the precise location of objects in space and time. All measurements of position, distance, or speed must be made with respect to a chosen reference frame.

• Example: A person walking at 2 m/s on a train moving at 25 m/s—speeds are relative to the chosen frame.

• Coordinate axes are often used to represent a reference frame.

Vectors and Scalars

Definitions and Examples

Physical quantities in physics are classified as either vectors or scalars:

• Vector: A quantity with both magnitude and direction (e.g., displacement, velocity, force).

• Scalar: A quantity with magnitude only, no direction (e.g., time, mass, volume).

Vectors are represented by arrows in diagrams; scalars are represented by numbers and units only.

Practice: Identifying Vectors and Scalars

• Acceleration of a plane as it takes off: Vector

• Number of passengers on the plane: Scalar

• Duration of the flight: Scalar

• Displacement of the flight: Vector

• Amount of fuel required: Scalar

Displacement and Distance

Definitions

• Displacement: The change in position of an object; a vector quantity.

• Distance travelled: The total length of the path taken between two positions; a scalar quantity.

• Distance cannot be negative; displacement can be positive or negative depending on direction.

Example: If you walk 2 m forward and then 2 m back to your starting point, your distance travelled is 4 m, but your displacement is 0 m.

Practice Problems

• A dog stands 2 m in front of you, you throw a ball 10 m, the dog retrieves and returns it directly. Displacement: 0 m (returns to start), Distance: 20 m (10 m out, 10 m back).

• A ball falls 0.1 m and returns: Displacement: 0 m, Distance: 0.2 m.

• Racer on a 500 m radius track, 4 laps: Displacement: 0 m (returns to start), Distance: m.

• City drive: 2 km north, 4 km east, 2 km south, 4 km east. Displacement: 8 km east, Distance: 12 km.

Velocity and Speed

Definitions

• Average velocity: Displacement divided by elapsed time.

• Average speed: Total distance travelled divided by elapsed time.

• Instantaneous velocity: Velocity at a specific instant.

• Instantaneous speed: Magnitude of instantaneous velocity.

• Velocity is a vector; speed is a scalar.

Practice Problems

• Walk with average velocity 0.98 m/s for 30 min: m.

• Bike at 6.5 m/s south for 90 s: m.

• Two students, 1.2 m/s and 1.5 m/s, 780 m: Time difference s.

Acceleration

Definitions and Types

• Acceleration: The rate at which velocity changes with time.

• Average acceleration: Change in velocity over a time interval.

• Instantaneous acceleration: Acceleration at a specific instant.

• Positive acceleration: Speed increases in the positive direction.

• Negative acceleration (deceleration): Speed decreases or increases in the negative direction.

Constant Acceleration

Kinematic Equations

For motion with constant acceleration, the following equations apply:

Problem-Solving Strategy:

1. Identify all given quantities.

2. Identify unknowns to be found.

3. Choose the appropriate equation.

4. Plug in values and solve.

Practice Problems

• Car accelerates at m/s2 from m/s to m/s: s.

• Skateboard accelerates from $0 m/s in s: m/s2, m.

• Aircraft liftoff: , m/s2.

Graphical Analysis of Linear Motion

Position-Time and Velocity-Time Graphs

• Position-time graph: Slope gives velocity.

• Concave up: Positive acceleration; concave down: Negative acceleration.

• Velocity-time graph: Slope gives acceleration.

Free Fall

Definition and Analysis

Free fall describes the motion of objects under the influence of gravity alone, with air resistance neglected. Near Earth's surface, the acceleration due to gravity is m/s2.

• All objects fall with the same acceleration regardless of mass.

• Equations of motion for free fall are the same as for constant acceleration, with .

Example: If a rock is thrown upward from a cliff with m/s, m/s2, and s, use to find the height.

Vector Addition and Resolution

Adding Vectors

• Resultant vector: The sum of two or more vectors.

• Graphical method: Place vectors tip-to-tail; the resultant is from the tail of the first to the tip of the last.

• Analytical method: Use components and the Pythagorean theorem.

For perpendicular vectors:

• Magnitude:

• Direction:

Vector Resolution

• Any vector can be resolved into x and y components: ,

• Sum all x-components and y-components to find the resultant's components.

• Use the Pythagorean theorem and inverse tangent to find magnitude and direction.

Example: A ball is shot at 8 m/s at 45°: m/s, m/s.

Projectile Motion

Types and Analysis

Projectile motion describes the motion of objects moving through the air under gravity, typically in two dimensions. Air resistance is neglected.

• Horizontal and vertical motions are analyzed separately.

• Horizontal motion: constant velocity ().

• Vertical motion: constant acceleration ().

• The path (trajectory) is parabolic.

Types of Projectile Motion

• Type 1: Launched horizontally from a height ().

• Type 2: Launched and lands at the same height ().

• Type 3: Launched and lands at different heights ().

Projectile Motion Equations

• Horizontal:

• Vertical:

Example: A rock is kicked horizontally from a 321 m high bridge, lands 45 m away. Find initial speed: , .

Relative Motion

Concept and Examples

Relative motion describes how the velocity of an object depends on the observer's frame of reference.

• Velocity of object relative to medium:

• Velocity of medium relative to observer:

• Velocity of object relative to observer:

Example: A rower paddles at 1 m/s against a 3 m/s current. Relative to shore: m/s (downstream).

Summary Table: Scalars vs. Vectors

Additional info: Some context and equations were inferred and expanded for completeness and clarity, as is standard in college-level physics study guides.