Motion in Two Dimensions: Kinematics and Circular Motion

Motion in Two Dimensions

Position and Displacement in the Plane

Motion in two dimensions involves tracking the position of an object as it moves along a path in the xy-plane. The position of a particle is described by a position vector \( \vec{r} \), which points from the origin to the particle's location.

• Position Vector: \( \vec{r} = x \hat{i} + y \hat{j} \), where x and y are the coordinates of the particle.

• Displacement: The change in position between two points is \( \Delta \vec{r} = \vec{r}_2 - \vec{r}_1 = \Delta x \hat{i} + \Delta y \hat{j} \).

Velocity in Two Dimensions

The average velocity is the displacement divided by the time interval, and the instantaneous velocity is the derivative of the position vector with respect to time.

• Average Velocity: \( \vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} = \frac{\Delta x}{\Delta t} \hat{i} + \frac{\Delta y}{\Delta t} \hat{j} \)

• Instantaneous Velocity: \( \vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt} = \frac{dx}{dt} \hat{i} + \frac{dy}{dt} \hat{j} \)

• The velocity vector is always tangent to the trajectory at any point.

Velocity Components and Direction

The velocity vector can be broken into x and y components, which are related to the speed and direction of motion.

• Given speed v and angle \( \theta \) (measured counterclockwise from the +x axis):

• \( v_x = v \cos \theta \)

• \( v_y = v \sin \theta \)

• \( v = \sqrt{v_x^2 + v_y^2} \)

• \( \theta = \tan^{-1} \left( \frac{v_y}{v_x} \right) \)

Acceleration in Two Dimensions

Acceleration is the rate of change of velocity. In two dimensions, acceleration can change the speed, the direction, or both.

• Average Acceleration: \( \vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} \)

• Instantaneous Acceleration: \( \vec{a} = \frac{d\vec{v}}{dt} = \frac{dv_x}{dt} \hat{i} + \frac{dv_y}{dt} \hat{j} \)

• Acceleration can be decomposed into two components:

  • \( \vec{a}_{\parallel} \): Parallel to velocity, changes speed.

  • \( \vec{a}_{\perp} \): Perpendicular to velocity, changes direction.

Finding the Acceleration Vector

To determine the acceleration between two velocity vectors:

1. Draw the velocity vectors with their tails together.

2. Draw the vector from the tip of the initial velocity to the tip of the final velocity; this is \( \Delta \vec{v} \).

3. Draw the average acceleration vector at the midpoint in the direction of \( \Delta \vec{v} \).

Acceleration Along a Curved Path

As an object moves along a curved path, the acceleration vector can have both parallel and perpendicular components relative to the velocity vector. The parallel component changes the speed, while the perpendicular component changes the direction.

Component Form of Acceleration

It is often convenient to express acceleration in terms of its x and y components:

• \( \vec{a} = a_x \hat{i} + a_y \hat{j} \)

• \( a_x = \frac{dv_x}{dt} \), \( a_y = \frac{dv_y}{dt} \)

Projectile Motion

Definition and Characteristics

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.

• Examples: thrown balls, diving athletes, objects in free fall with horizontal velocity.

• The horizontal and vertical motions are independent except for sharing the same time interval.

Projectile Launch and Initial Velocity

The initial velocity of a projectile can be decomposed into horizontal and vertical components based on the launch angle \( \theta \):

• \( v_{0x} = v_0 \cos \theta \)

• \( v_{0y} = v_0 \sin \theta \)

• Acceleration: \( a_x = 0 \), \( a_y = -g \)

Projectile Motion Example

For a projectile launched with \( \vec{v}_0 = (9.8 \hat{i} + 19.6 \hat{j}) \) m/s:

• Horizontal velocity \( v_x \) remains constant.

• Vertical velocity \( v_y \) decreases by 9.8 m/s each second due to gravity.

• Final velocity at landing: \( \vec{v}_f = 9.8 \hat{i} - 19.6 \hat{j} \) m/s.

Independence of Motion

The horizontal and vertical motions of a projectile are independent. Objects dropped and objects projected horizontally from the same height hit the ground at the same time if released simultaneously.

Conceptual Understanding: The Falling Coconut

If an arrow is aimed directly at a coconut that falls the instant the arrow is released, the arrow will hit the coconut. Both the arrow and coconut experience the same vertical acceleration due to gravity, so their vertical displacements are identical over time.

Kinematic Equations for Projectile Motion

The equations for projectile motion are derived from the kinematic equations for constant acceleration, applied separately to the horizontal and vertical directions:

Horizontal	Vertical
\( x_f = x_i + v_{ix} \Delta t \)	\( y_f = y_i + v_{iy} \Delta t - \frac{1}{2} g (\Delta t)^2 \)
\( v_{fx} = v_{ix} = \text{constant} \)	\( v_{fy} = v_{iy} - g \Delta t \)

Projectile Range and Launch Angle

The range of a projectile (horizontal distance traveled) depends on the initial speed and launch angle. Maximum range is achieved at a 45° launch angle, and the range is proportional to \( v_0^2 \sin 2\theta \).

Relative Motion and Reference Frames

Relative Velocity

The velocity of an object depends on the observer's frame of reference. The relative velocity is the velocity of one object as measured from another moving reference frame.

• Notation: \( v_{AB} \) is the velocity of A relative to B.

Reference Frames

A reference frame is a coordinate system in which position and time are measured. Transformations between reference frames are necessary to compare measurements made by different observers.

Circular Motion

Uniform Circular Motion

Uniform circular motion describes the motion of a particle traveling at constant speed along a circular path of radius r.

• Velocity is always tangent to the circle.

• The period T is the time for one complete revolution: \( v = \frac{2\pi r}{T} \)

Angular Position, Displacement, and Velocity

The angular position \( \theta \) describes the location of a particle on a circle. The angular displacement is \( \Delta \theta = \theta_f - \theta_i \), and the angular velocity is the rate of change of angular position:

• \( \omega = \frac{d\theta}{dt} \)

• Units: radians per second (rad/s)

Tangential Velocity and Centripetal Acceleration

The tangential velocity is related to angular velocity by \( v_t = \omega r \). In uniform circular motion, the acceleration is always directed toward the center of the circle (centripetal acceleration):

• \( a_c = \frac{v^2}{r} = \omega^2 r \)

Summary Table: Uniform Circular Motion Model

Quantity	Expression
Tangential velocity	\( v_t = \omega r \)
Centripetal acceleration	\( a_c = \frac{v_t^2}{r} = \omega^2 r \)
Period	\( T = \frac{2\pi}{\omega} \)

Nonuniform Circular Motion and Angular Acceleration

When the angular velocity changes with time, the motion is nonuniform. The angular acceleration \( \alpha \) is defined as:

• \( \alpha = \frac{d\omega}{dt} \)

• Units: rad/s2

• Sign convention: Positive if angular velocity increases counterclockwise, negative if clockwise.

Tangential Acceleration

When an object speeds up or slows down in circular motion, it experiences tangential acceleration:

• \( a_t = \frac{dv_t}{dt} = r \alpha \)

• Total acceleration is the vector sum of tangential and centripetal accelerations.