Motion in a Plane

Introduction to Two-Dimensional Motion

Motion in a plane, or two-dimensional motion, is a fundamental topic in college physics. It involves analyzing the position, velocity, and acceleration of objects moving in two spatial dimensions, typically represented by the x- and y-axes. This topic is essential for understanding projectile motion, relative velocity, and the application of vector mathematics in physics.

• Position Vector: Describes the location of a particle in the plane relative to the origin.

• Velocity Vector: Represents the rate of change of position with respect to time.

• Acceleration Vector: Represents the rate of change of velocity with respect to time.

Goals for Chapter 3

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

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

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

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

Position, Displacement, and Vectors in 2D

Position Vector and Magnitude

The position vector \( \vec{r} \) locates a point P in the x-y plane relative to the origin. The magnitude of the position vector gives the distance from the origin to point P:

• Position vector: \( \vec{r} = x\hat{i} + y\hat{j} \)

• Magnitude: \( r = |\vec{r}| = \sqrt{x^2 + y^2} \)

Displacement is the change in position vector:

• \( \Delta \vec{r} = \vec{r}_2 - \vec{r}_1 \)

Vector Representation: Cartesian and Polar Coordinates

• Vectors can be expressed in terms of their Cartesian components (x, y) or as magnitude and angle (polar form).

• Conversion between forms:

  • \( x = r \cos \theta \)

  • \( y = r \sin \theta \)

  • \( r = \sqrt{x^2 + y^2} \)

  • \( \theta = \tan^{-1}(y/x) \)

Velocity in a Plane

Average and Instantaneous Velocity

Velocity in two dimensions is a vector quantity describing both the speed and direction of motion. It can be analyzed as:

• Average velocity: The total displacement divided by the total time interval.

  • \( \vec{v}_{\text{av}} = \frac{\Delta \vec{r}}{\Delta t} \)

• Instantaneous velocity: The velocity at a specific instant, tangent to the path.

  • \( \vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} \)

Key Point: The instantaneous velocity vector is always tangent to the object's path in the x-y plane.

Example: Motion of a Model Car

Consider a car moving from point P1 to P2 in the x-y plane:

• \( \Delta x = 3.0\,\text{m} \), \( \Delta y = 4.0\,\text{m} \), \( \Delta t = 0.5\,\text{s} \)

• Average velocity components:

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

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

• Magnitude of average velocity: \( v_{\text{av}} = \sqrt{v_{\text{av},x}^2 + v_{\text{av},y}^2} \)

Acceleration in a Plane

Average and Instantaneous Acceleration

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

• Average acceleration: The change in velocity divided by the time interval.

  • \( \vec{a}_{\text{av}} = \frac{\Delta \vec{v}}{\Delta t} \)

• Instantaneous acceleration: The acceleration at a specific instant.

  • \( \vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} \)

Key Point: The acceleration vector always points toward the concave side of the curved path.

Vector Addition and Subtraction

• Addition (Head-to-Tail Method): 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.

• Subtraction: Add the opposite of the vector to be subtracted (reverse its direction by 180°).

Projectile Motion

Characteristics of Projectile Motion

Projectile motion is a special case of two-dimensional motion where an object moves under the influence of gravity alone (neglecting air resistance). The path followed is a parabola in the x-y plane.

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

• Horizontal motion: constant velocity (no horizontal acceleration).

• Vertical motion: constant acceleration due to gravity (\( a_y = -g \)).

Equations of Motion for Projectiles

Initial Velocity Components

• \( v_{0x} = v_0 \cos \theta_0 \)

• \( v_{0y} = v_0 \sin \theta_0 \)

Key Quantities in Projectile Motion

• Maximum height: The highest vertical position reached by the projectile.

• Range: The horizontal distance traveled by the projectile.

• Time of flight: The total time the projectile is in the air.

For a projectile launched from ground level (\( y_0 = 0 \)):

• Time of flight: \( T = \frac{2 v_0 \sin \theta_0}{g} \)

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

• Maximum height: \( H = \frac{v_0^2 \sin^2 \theta_0}{2g} \)

Examples and Applications

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

• Engineering: Determining the range and height for projectiles in design problems.

Summary Table: Key Equations for Projectile Motion

Additional info:

• Relative velocity in two dimensions and frames of reference are also important but not detailed in the provided slides. In general, relative velocity is calculated by vector addition or subtraction of velocities as observed from different frames.