Motion in Two Dimensions

Learning Goals

This chapter introduces the analysis of motion in two or three dimensions using vectors. The main objectives are:

• Representing position and velocity of a particle using vectors in two or three dimensions.

• Finding vector acceleration and interpreting its components parallel and perpendicular to a particle’s path.

• Solving problems involving curved paths, such as projectile motion.

• Analyzing circular motion with constant or varying speed.

• Relating velocities as seen from different frames of reference.

Position and Displacement in 2D

Position Vectors

In two dimensions, the position of a particle is represented by a vector from the origin to the particle’s location.

• The position vector r at time t has components (x, y).

• Displacement is the change in position, given by the vector difference:

• Components: ,

Velocity in Two Dimensions

Average and Instantaneous Velocity

Velocity is the rate of change of position. In two dimensions:

• Average velocity:

• Instantaneous velocity:

• The velocity vector is always tangent to the path of the particle.

Example: If a comet moves from (2.0 m, 1.0 m) to (3.0 m, 5.0 m) in 4.0 s, its average velocity is:

Acceleration in Two Dimensions

Average and Instantaneous Acceleration

Acceleration describes how velocity changes with time.

• Average acceleration:

• Instantaneous acceleration:

• The direction of average acceleration is the same as the change in velocity .

Components of Acceleration

• Acceleration can be decomposed into components parallel () and perpendicular () to the velocity vector.

• changes the speed; changes the direction.

• For curved paths, points toward the center of curvature (concave side).

Motion in a Circular Path

Uniform Circular Motion

When a particle moves in a circle at constant speed:

• The acceleration is always directed toward the center (centripetal acceleration).

• Magnitude: , where is speed and is the radius.

• The period (time for one revolution):

• Velocity and acceleration vectors are always perpendicular.

Nonuniform Circular Motion

• If speed varies, there is a tangential acceleration in addition to the radial (centripetal) acceleration .

• (radial), (tangential)

• The total acceleration is the vector sum of and .

Projectile Motion

Basic Principles

A projectile is any object given an initial velocity and then allowed to move under the influence of gravity alone (neglecting air resistance).

• Projectile motion occurs in a vertical plane.

• The horizontal motion has constant velocity; the vertical motion has constant acceleration .

• At the peak of the trajectory, velocity and acceleration are perpendicular.

Equations of Projectile Motion

• Let initial position be , initial speed , and launch angle .

• Horizontal motion:

• Vertical motion:

• Horizontal velocity: (constant)

• Vertical velocity:

• The trajectory is a parabola:

Projectile Range

• The range (horizontal distance traveled) is maximized for a launch angle of (on level ground).

• Range formula:

• On the Moon, where is smaller, the range increases (for the same and ).

Relative Motion

Frames of Reference

The velocity of an object depends on the observer’s frame of reference.

• Relative velocity:

• Example: A person walks at 1.0 m/s on a train moving at 10 m/s. Relative to the ground, their speed is 11 m/s (if walking in the same direction as the train).

Applications

• River crossing problems: Swimmers or boats must account for both their own speed and the current’s speed.

• To move directly across a river, a swimmer must aim upstream to compensate for the current.

Effects of Air Resistance

Air Drag in Projectile Motion

When air resistance is significant:

• Acceleration is no longer constant.

• The trajectory is not a perfect parabola; maximum height and range decrease.

• Air drag acts opposite to the velocity vector.

Direction of Drag Force

• The acceleration due to drag always points opposite to the direction of motion.

• After the peak of a projectile’s flight, drag continues to oppose the velocity, affecting both horizontal and vertical components.

Summary Table: Key Equations for 2D Motion

Additional info: Some context and equations have been inferred and expanded for clarity and completeness, based on standard college physics curriculum.