Motion in Two or Three Dimensions: Study Notes

Motion in Two or Three Dimensions

Introduction

Many real-world motions, such as the flight of a baseball, the path of a roller coaster, or the flight of a hawk, occur in two or three dimensions. Understanding these motions requires extending the concepts of kinematics from one dimension to multiple dimensions.

• Key Question: How do we describe and predict the motion of objects moving along curved paths?

• Applications: Sports, amusement park rides, and animal flight patterns.

Position, Velocity, and Acceleration in Multiple Dimensions

Position Vector

The position vector locates a particle in space relative to an origin. In three dimensions, it is expressed as:

• Definition: The position vector from the origin to point P has components x, y, and z.

$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$

Velocity

Velocity describes how the position of a particle changes with time.

• Average velocity: Displacement divided by the time interval:

$\vec{v}_{\text{av}} = \frac{\Delta \vec{r}}{\Delta t}$

• Instantaneous velocity: The rate of change of position with respect to time:

$\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$

• The instantaneous velocity vector is always tangent to the particle's path.

Average Velocity

The average velocity between two points is the displacement divided by the time interval between those points. It points in the same direction as the displacement vector.

• Graphically, the displacement vector connects the initial and final positions, and the average velocity is parallel to this vector.

Instantaneous Velocity

• The components of instantaneous velocity are:

$v_x = \frac{dx}{dt},\quad v_y = \frac{dy}{dt},\quad v_z = \frac{dz}{dt}$

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

Acceleration

Acceleration describes how the velocity of a particle changes with time.

• It can change in magnitude, direction, or both.

Average Acceleration

• Defined as the change in velocity divided by the time interval:

$\vec{a}_{\text{av}} = \frac{\Delta \vec{v}}{\Delta t}$

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

Instantaneous Acceleration

• The rate of change of velocity with respect to time:

$\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$

• The acceleration vector does not have to be tangent to the path; for curved paths, it points toward the concave side.

Components of Acceleration

• Each component is the rate of change of the corresponding velocity component:

$a_x = \frac{dv_x}{dt},\quad a_y = \frac{dv_y}{dt},\quad a_z = \frac{dv_z}{dt}$

• Example: Shooting an arrow involves both x- and y-components of acceleration.

Parallel and Perpendicular Components of Acceleration

• For a particle moving along a curved path:

• With constant speed, acceleration is normal (perpendicular) to the path.

• With increasing speed, acceleration points ahead of the normal.

• With decreasing speed, acceleration points behind the normal.

Projectile Motion

Definition and Assumptions

• A projectile is any object given an initial velocity and then allowed to move under the influence of gravity and air resistance (often neglected in basic analysis).

• Neglect air resistance and Earth's curvature for introductory problems.

The X- and Y-Motion Are Separable

• Projectile motion can be analyzed as:

  • Horizontal motion with constant velocity ($a_x = 0$)

  • Vertical motion with constant acceleration ($a_y = -g$)

• Example: Dropping and throwing balls horizontally from the same height—both hit the ground at the same time.

Equations for Projectile Motion

• If the initial position is $x_0 = 0$, $y_0 = 0$, the equations are:

$x = v_0 \cos \theta_0 \; t$

$y = v_0 \sin \theta_0 \; t - \frac{1}{2}gt^2$

$v_x = v_0 \cos \theta_0$

$v_y = v_0 \sin \theta_0 - gt$

• These equations describe the position and velocity of a projectile at any time t.

Projectile Motion – Initial Velocity

• The initial velocity components are determined by the initial speed and launch angle:

$v_{0x} = v_0 \cos \theta_0$

$v_{0y} = v_0 \sin \theta_0$

• Example: A soccer ball kicked at an angle has both horizontal and vertical velocity components.

Parabolic Trajectories

• The path of a projectile (neglecting air resistance) is a parabola.

• Example: A bouncing ball loses energy with each bounce, resulting in successively lower peaks.

Motion in a Circle

Uniform Circular Motion

Uniform circular motion occurs when an object moves at constant speed along a circular path.

• Acceleration is always perpendicular to velocity (no parallel component).

• Acceleration points toward the center of the circle (centripetal acceleration).

$a_{\text{rad}} = \frac{v^2}{R}$

• The period $T$ (time for one revolution):

$a_{\text{rad}} = \frac{4\pi^2 R}{T^2}$

• Velocity and acceleration vectors are always perpendicular.

Nonuniform Circular Motion

• If the speed varies, the motion is nonuniform circular motion.

• There is a radial (centripetal) acceleration $a_{\text{rad}} = v^2 / R$ and a tangential acceleration $a_{\text{tan}}$ parallel to the instantaneous velocity.

Relative Velocity

Definition and Frame of Reference

• The velocity of an object as seen by a particular observer is called relative velocity.

• A frame of reference consists of a coordinate system and a time scale.

• Relative velocity is crucial in many practical situations (e.g., moving vehicles, aircraft, boats).

Relative Velocity in One Dimension

• If point P moves relative to frame A, its velocity is $v_{P/A}$.

• If P moves relative to frame B, and B moves relative to A, then:

$v_{P/A} = v_{P/B} + v_{B/A}$

• This equation allows us to relate velocities measured in different reference frames.

• Example: A passenger walking inside a moving train—her velocity relative to the ground is the sum of her velocity relative to the train and the train's velocity relative to the ground.

Additional info: These notes are based on slides from "University Physics with Modern Physics, 15th Edition, Chapter 3: Motion in Two or Three Dimensions." All equations are standard results in introductory physics. Examples and applications are inferred from the context of the slides and standard textbook treatments.