Kinematics in Two Dimensions: Kinematics, Projectile Motion, and Angular Motion (Physics 130, Lecture 4)

Kinematics in Two Dimensions

Introduction to 2D Kinematics

Kinematics in two dimensions extends the study of motion to objects moving in a plane, requiring the analysis of both x and y components. This includes straight-line motion, projectile motion, and circular motion, all of which are foundational in physics.

• Displacement, velocity, and acceleration are all vector quantities and must be described in terms of their components.

• Motion can be decomposed into independent motions along perpendicular axes (usually x and y).

Acceleration in 2D

Vector Nature of Acceleration

Acceleration in two dimensions can change the magnitude and/or direction of velocity. The average acceleration is defined as:

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

• Perpendicular acceleration changes the direction of velocity (e.g., uniform circular motion).

• Parallel acceleration changes the magnitude of velocity.

• In general, acceleration can have both perpendicular and parallel components relative to the velocity vector.

Vectors in 2D Motion

Decomposing Vectors

Any vector in two dimensions can be decomposed into x and y components using trigonometry:

• $v_x = v \cos(\theta)$

• $v_y = v \sin(\theta)$

• $v = \sqrt{v_x^2 + v_y^2}$

• $\tan(\theta) = \frac{v_y}{v_x}$

Example: A hockey puck has $v = 2.0$ m/s at an angle of $30^\circ$.

• $v_x = 2.0 \cos 30^\circ = 1.7$ m/s

• $v_y = 2.0 \sin 30^\circ = 1.0$ m/s

Uniform Motion in 2D

Constant Velocity Motion

For an object moving at constant velocity (no acceleration), the position as a function of time is:

• $x = v_x t + x_0$

• $y = v_y t + y_0$

• The total displacement $d$ from the origin after time $t$ is $d = \sqrt{x^2 + y^2}$

• Alternatively, $d = v t$ if the motion is in a straight line.

Example: A puck starts at $(0,0)$ with $v_x = 2$ m/s, $v_y = 1$ m/s. After $t = 5$ s:

• $x = 2 \times 5 = 10$ m

• $y = 1 \times 5 = 5$ m

• $d = \sqrt{10^2 + 5^2} = 11.2$ m$

Motion with Constant Acceleration

Kinematic Equations in 2D

When acceleration is present, the kinematic equations must be applied to each component:

• $x = x_0 + v_{x0} t + \frac{1}{2} a_x t^2$

• $v_x = v_{x0} + a_x t$

• $v_x^2 = v_{x0}^2 + 2 a_x (x - x_0)$

• Similarly for y: $y = y_0 + v_{y0} t + \frac{1}{2} a_y t^2$, etc.

Example: A puck at rest receives $\vec{a} = 10\, (m/s^2) \hat{x} + 20\, (m/s^2) \hat{y}$ for $0.10$ s.

• $v_{fx} = 10 \times 0.1 = 1$ m/s

• $v_{fy} = 20 \times 0.1 = 2$ m/s

• $|\vec{v}_f| = \sqrt{1^2 + 2^2} = 2.24$ m/s

• Angle: $\theta = \tan^{-1}(2/1) = 63.4^\circ$

Free Fall and Vertical Motion

Falling Objects: Upward and Downward

Objects in free fall experience constant acceleration due to gravity ($g = 9.8$ m/s2 downward). The motion can be analyzed for both upward and downward trajectories.

Case	Position	Fall Time	Velocity	Acceleration
Falling Downward	Decreases	Increases	Becomes more negative	-9.8 m/s2
Falling Upward	Increases then decreases	Increases	Decreases to zero at top, then negative	-9.8 m/s2

• At maximum height, velocity is zero; acceleration remains $-9.8$ m/s2.

• Time to rise equals time to fall (if starting and ending at same altitude).

Projectile Motion

Basic Principles

Projectile motion describes the motion of an object launched into the air, subject only to gravity (neglecting air resistance). The motion is a combination of uniform motion in the horizontal direction and uniformly accelerated motion in the vertical direction.

• Horizontal motion: $a_x = 0$, $v_x = v_0 \cos \theta$ (constant)

• Vertical motion: $a_y = -g$, $v_y = v_0 \sin \theta - g t$

• Trajectory is parabolic.

Key Equations:

• Time of flight (for same start/end altitude): $t = \frac{2 v_0 \sin \theta}{g}$

• Range: $R = \frac{v_0^2 \sin 2\theta}{g}$

• Maximum height: $h_{\text{max}} = \frac{(v_0 \sin \theta)^2}{2g}$

Example: A projectile is fired at $v_0 = 100$ m/s, $\theta = 30^\circ$.

• Time in air: $t = \frac{2 \times 100 \sin 30^\circ}{9.8} = 10.2$ s

• Range: $R = \frac{100^2 \sin 60^\circ}{9.8} = 883$ m

Special Cases and Conceptual Questions

Comparing Upward and Downward Throws

If two balls are thrown from a cliff, one upward and one downward with the same speed, both will have the same speed upon reaching the ground (neglecting air resistance), but the upward-thrown ball takes longer to fall.

• Final speed depends only on initial height and speed, not direction.

• Time to fall is longer for the upward-thrown ball.

Summary Table: Free Fall Motion

Case	Velocity	Acceleration
Falling Downward	Increasing (negative)	Constant, downward
Falling Upward	Decreasing (positive to zero to negative)	Constant, downward

Additional info:

• Further topics such as angular motion, centripetal acceleration, and more advanced projectile motion are likely covered in subsequent slides or lectures.

• Students should be familiar with vector decomposition, kinematic equations, and the independence of horizontal and vertical motions in projectile problems.