Kinematics and Motion in One and Two Dimensions

Finding Position from Velocity

Kinematics is the study of motion without considering its causes. One of the fundamental problems in kinematics is determining an object's position when its velocity as a function of time is known.

• Position Function: If the velocity is known, the position at time can be found from the initial position at time by integrating the velocity:

• Integral Interpretation: The integral represents the total area under the velocity vs. time curve between and .

• Displacement: The total displacement is the area under the velocity curve.

• Graphical Representation: The area under the curve can be approximated by dividing the time interval into small steps where velocity is nearly constant, summing the areas of rectangles, and taking the limit as the step size approaches zero.

Example: If a car's velocity increases linearly from 0 to 16 m/s over 3 seconds, the displacement is the area of a triangle under the velocity-time graph: .

Motion on an Inclined Plane

When an object moves on an inclined plane, gravity causes acceleration along the slope.

• Acceleration Along the Incline:

• Direction: The sign of depends on the direction of the incline and the motion.

• Free-Body Diagram: The gravitational force can be decomposed into components parallel and perpendicular to the incline.

Example: A ball rolling up and then down a ramp will have acceleration graphs showing zero acceleration on flat segments and constant acceleration (positive or negative) on the slope.

Position, Velocity, and Acceleration Graphs

Understanding the relationships between position, velocity, and acceleration is crucial in kinematics.

• Instantaneous Velocity: (slope of the position-time graph)

• Instantaneous Acceleration: (slope of the velocity-time graph)

• Displacement: Area under the velocity-time graph

• Velocity: Area under the acceleration-time graph

Example: For a ball on a track with flat and inclined segments, the position graph is piecewise smooth, the velocity graph is constant on flats and linear on slopes, and the acceleration graph is zero on flats and constant on slopes.

Equations of Motion for Constant Acceleration

For motion with constant acceleration, the following kinematic equations apply:

Example: A springbok leaps upward, accelerating at for ; use the equations above to find the maximum height reached.

Vectors and Vector Operations

Scalars and Vectors

Physical quantities can be classified as scalars or vectors.

• Scalar: A quantity described by magnitude only (e.g., mass, temperature).

• Vector: A quantity described by both magnitude and direction (e.g., displacement, velocity, acceleration).

• Vector Notation: Vectors are represented by arrows; the length indicates magnitude, and the arrowhead indicates direction. Common symbols: (position), (velocity), (acceleration).

Vector Addition and Subtraction

Vectors can be added graphically or analytically.

• Tip-to-Tail Rule: Place the tail of one vector at the tip of another; the resultant vector is drawn from the tail of the first to the tip of the last.

• Parallelogram Rule: Place both vectors at a common origin; the diagonal of the parallelogram represents the sum.

• Magnitude of Resultant: For vectors at right angles:

• Direction of Resultant:

Example: A bird flies 100 m east, then 50 m northwest (45° north of west). The net displacement is found by vector addition, using graphical or component methods.

Vector Components and Unit Vectors

Any vector in a plane can be decomposed into components along the x- and y-axes.

• Component Form:

• Finding Components: , (where is the angle from the x-axis)

• Magnitude from Components:

• Direction from Components:

• Unit Vectors: (x-direction), (y-direction); both have magnitude 1.

Example: A vector with components , has magnitude and direction above the x-axis.

Summary Table: Kinematic Quantities and Their Relationships

Key Points and Concepts

• The sign of velocity indicates direction (positive: right/up, negative: left/down).

• The sign of acceleration indicates the direction of the acceleration vector, not whether the object is speeding up or slowing down.

• An object speeds up if velocity and acceleration have the same sign; it slows down if they have opposite signs.

• Vectors are equal if they have the same magnitude and direction, regardless of their initial points.

• Coordinate systems are chosen for convenience; axes can be oriented as needed.

Additional info: These notes synthesize and expand upon the provided slides and images, filling in standard academic context for introductory college physics on kinematics and vectors.