Motion and Kinematics

Position Vector

The position vector describes the location of a particle in space relative to an origin. It is fundamental in representing motion in one, two, or three dimensions.

• Definition: The position vector \( \vec{r} \) has components along the x, y, and z axes:

• Application: Used to specify the exact location of a particle at any given time.

• Example: If a particle is at coordinates (2, 3, 5), its position vector is .

Displacement Vector

The displacement vector represents the change in position of a particle between two points in space.

• Definition:

• Key Point: Displacement is a vector quantity, indicating both magnitude and direction.

• Example: If a particle moves from (1, 2, 3) to (4, 6, 8), the displacement is .

Average Velocity

Average velocity is the displacement divided by the time interval over which the displacement occurs. It is a vector quantity.

• Formula:

• Direction: Same as the direction of displacement.

• Example: If a dragster moves from m at s to m at s, m, s, so m/s.

Instantaneous Velocity

Instantaneous velocity is the rate of change of position at a specific instant. It is the derivative of the position vector with respect to time.

• Formula:

• Component Form:

• Example: For ,

Position-Time (x-t) Graphs

An x-t graph plots position as a function of time. The slope of the graph at any point gives the velocity.

• Average velocity: Slope of the line joining two points.

• Instantaneous velocity: Slope of the tangent at a point.

• Speed: Steeper slope indicates greater speed.

Worked Example: Mars Rover Motion

Consider a Mars rover with position given by and .

• Find: Coordinates and distance at s, displacement and average velocity for to $t = 2.0$ s, and instantaneous velocity at $t = 2.0$ s.

• Application: Illustrates calculation of vector quantities in kinematics.

Acceleration

Acceleration Vector

Acceleration is the rate of change of velocity. It is a vector quantity and can be decomposed into components.

• Average acceleration:

• Instantaneous acceleration:

• Direction: Same as the change in velocity.

Acceleration Example: Jet Plane

Given velocity components at two times, calculate average acceleration and its direction.

• Application: Useful for analyzing motion in two dimensions.

Acceleration Components: Tangential and Perpendicular

The acceleration vector can be split into components parallel and perpendicular to the velocity vector.

• Tangential component: Changes the speed.

• Perpendicular component: Changes the direction of motion.

Motion with Constant Acceleration

Kinematic Equations

For motion with constant acceleration, the following equations apply:

Projectile Motion

Projectile Trajectory

A projectile follows a curved path under the influence of gravity. Its motion can be analyzed in horizontal and vertical components.

• Horizontal motion: Constant velocity,

• Vertical motion: Constant acceleration,

• Equations:

Projectile Motion Example: Different Initial and Final Heights

Calculate the horizontal distance a ball travels when thrown from a window above the ground, given initial velocity and angle.

• Application: Demonstrates use of kinematic equations and quadratic formula for time of flight.

Motion in a Circle

Uniform and Non-Uniform Circular Motion

Motion in a circle can be uniform (constant speed) or non-uniform (changing speed). The acceleration in circular motion is called centripetal acceleration.

• Radial acceleration:

• Period: Time for one revolution,

• Frequency:

• Angular speed:

• Speed:

Centripetal Acceleration

In uniform circular motion, the acceleration is always directed toward the center of the circle.

• Formula:

• Direction: Perpendicular to velocity, toward the center.

Relative Velocity

Frames of Reference

Relative velocity describes how the velocity of an object appears from different frames of reference.

• Formula:

• Application: Used in problems involving moving observers, such as trains, cars, or airplanes.

Summary Table: Kinematic Quantities

Additional info: These notes cover the fundamental concepts of kinematics, including vectors, motion in one and two dimensions, projectile motion, circular motion, and relative velocity, as outlined in chapters 2 and 3 of a typical college physics textbook.