Motion in a Plane: Two-Dimensional Kinematics

Chapter 3: Motion in a Plane

Introduction

This chapter explores the physics of motion in two dimensions, focusing on vectors, projectile motion, and the analysis of position, velocity, and acceleration in a plane. Understanding these concepts is essential for describing the motion of objects such as projectiles, vehicles, and particles in a variety of physical contexts.

Goals for Chapter 3

• To study and calculate position, velocity, and acceleration vectors in 2D.

• To frame two-dimensional motion as it occurs in the motion of projectiles.

• To use the equations of motion for constant acceleration to solve for unknown quantities for an object moving under constant acceleration in 2D.

• To study the relative velocity of an object for observers in different frames of reference in 2D.

Vectors in Two Dimensions

Position Vectors

The position vector describes the location of a point or particle in space relative to an origin. In two dimensions, it can be expressed in terms of its Cartesian components or in terms of magnitude and direction.

• Cartesian Components: The position vector \( \vec{r} \) has components \( x \) and \( y \) along the x- and y-axes, respectively.

• Magnitude of a Vector: The distance of point P from the origin is given by: $r = |\vec{r}| = \sqrt{x^2 + y^2}$

• Direction: The angle \( \theta \) the vector makes with the x-axis can be found using: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

Velocity in a Plane

Velocity in two dimensions is a vector quantity, having both magnitude and direction. It can be described as the rate of change of the position vector with respect to time.

• Average Velocity: The average velocity over a displacement \( \Delta \vec{r} \) during a time interval \( \Delta t \) is: $\vec{v}_{\text{av}} = \frac{\Delta \vec{r}}{\Delta t}$

• Instantaneous Velocity: The instantaneous velocity at a point is the derivative of the position vector with respect to time: $\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$

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

Example: Motion of a Model Car

Consider a car moving from point P_1 to P_2 in the xy-plane. The average velocity components can be calculated as:

• $v_{\text{av},x} = \frac{\Delta x}{\Delta t}$

• $v_{\text{av},y} = \frac{\Delta y}{\Delta t}$

For example, if \( \Delta x = 3.0 \; \text{m} \), \( \Delta y = 4.0 \; \text{m} \), and \( \Delta t = 2.5 \; \text{s} - 2.0 \; \text{s} = 0.5 \; \text{s} \):

• $v_{\text{av},x} = \frac{3.0}{0.5} = 6.0 \; \text{m/s}$

• $v_{\text{av},y} = \frac{4.0}{0.5} = 8.0 \; \text{m/s}$

Acceleration in a Plane

Definition and Calculation

Acceleration in two dimensions must account for changes in both the magnitude and direction of velocity.

• Average Acceleration: The average acceleration over a time interval \( \Delta t \) is: $\vec{a}_{\text{av}} = \frac{\Delta \vec{v}}{\Delta t}$

• Instantaneous Acceleration: The instantaneous acceleration at a point is: $\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$

• Acceleration always points toward the concave side of a curved path.

Vector Addition and Subtraction

Vectors in two dimensions can be added or subtracted using graphical or analytical methods.

• Head-to-Tail Method: To add two vectors, place the tail of the second vector at the head of the first. The resultant vector is drawn from the tail of the first to the head of the second.

• Component Method: Add the corresponding x- and y-components of the vectors.

• Subtraction: Subtracting a vector is equivalent to adding its opposite (reverse direction).

Projectile Motion

Characteristics of Projectile Motion

Projectile motion is a form of two-dimensional motion under constant acceleration due to gravity. The path followed by a projectile is a parabola in the absence of air resistance.

• The motion can be analyzed by separating it into horizontal (x) and vertical (y) components.

• Horizontal Motion: Constant velocity (no horizontal acceleration if air resistance is neglected).

• Vertical Motion: Constant acceleration due to gravity (g).

Equations of Motion for Projectiles

Quantity	X-Direction	Y-Direction
Position	$x(t) = x_0 + v_{0x} t$	$y(t) = y_0 + v_{0y} t - \frac{1}{2} g t^2$
Velocity	$v_x = v_{0x}$	$v_y = v_{0y} - g t$
Acceleration	$a_x = 0$	$a_y = -g$

Initial Velocity Components

• Given an initial speed v_0 at an angle \theta_0 above the horizontal:

  • $v_{0x} = v_0 \cos \theta_0$

  • $v_{0y} = v_0 \sin \theta_0$

Key Quantities in Projectile Motion

• Time of Flight (for y0 = 0): $t = \frac{2 v_{0y}}{g} = \frac{2 v_0 \sin \theta_0}{g}$

• Maximum Height: $h_{\text{max}} = \frac{v_{0y}^2}{2g} = \frac{(v_0 \sin \theta_0)^2}{2g}$

• Range (Horizontal Distance): $R = \frac{v_0^2 \sin 2\theta_0}{g}$

Example Applications

• Sports: Calculating the trajectory of a baseball, football, or basketball.

• Engineering: Predicting the path of projectiles in ballistics or fireworks.

Summary Table: Projectile Motion Quantities

Quantity	Formula	Description
Initial velocity (x-component)	$v_{0x} = v_0 \cos \theta_0$	Horizontal component of initial velocity
Initial velocity (y-component)	$v_{0y} = v_0 \sin \theta_0$	Vertical component of initial velocity
Time of flight	$t = \frac{2 v_{0y}}{g}$	Total time projectile is in the air
Maximum height	$h_{\text{max}} = \frac{v_{0y}^2}{2g}$	Highest vertical position reached
Range	$R = \frac{v_0^2 \sin 2\theta_0}{g}$	Horizontal distance traveled

Additional info:

• Projectile motion assumes air resistance is negligible unless otherwise stated.

• All equations assume the acceleration due to gravity g is constant and directed downward.