Quantities of 2D Motion and Projectile Motion – Study Notes

Chapter 3: Quantities of 2D Motion and Projectile Motion

Introduction

This chapter explores the fundamental concepts of motion in two dimensions, focusing on the analysis of projectile motion. The principles extend the ideas of 1D kinematics to 2D, introducing vector notation and the independence of horizontal and vertical motion.

Review of 1D Motion

1D Motion with Constant Velocity

• Equation of motion: Describes the position of an object moving at constant velocity.

• Formula: $x(t) = x_0 + v_x t$

• Key Point: Only initial position and velocity are needed to describe the motion.

1D Motion with Constant Acceleration

• Equations of motion: Used when acceleration is constant.

• Formulas: $v_x(t) = v_{0x} + a_x t$ $x(t) = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$

• Key Point: Requires initial position, initial velocity, and constant acceleration.

Free Fall in 1D

Gravitational Acceleration

• Definition: Free fall is motion under the influence of gravity alone.

• Near Earth's surface: All objects experience a constant downward acceleration $g = 9.8\ \mathrm{m/s^2}$.

• Equations for vertical motion (y-axis): $v_y = v_{0y} + a_y t$ $y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2$ $v_y^2 = v_{0y}^2 + 2 a_y \Delta y$ Where $a_y = -g$ (downward direction).

Example: Ball Thrown Upward

• Problem: A ball is thrown upward at $15.0\ \mathrm{m/s}$ from a $50.0\ \mathrm{m}$ high building.

• Find: Maximum height and time to hit the ground.

• Solution (using $v_y = 0$ at max height): $t_{max} = \frac{v_{0y}}{g} = \frac{15.0}{9.8} = 1.53\ \mathrm{s}$ $y_{max} = 50 + (15.0)(1.53) + \frac{1}{2}(-9.8)(1.53)^2 = 61.5\ \mathrm{m}$

• Time to hit ground: Use quadratic formula with $y = 0$.

2D Motion: Vectors and Kinematics

Position Vector and Displacement

• Position Vector ($\vec{r}$): Points from the origin to the object's location. $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$

• Displacement ($\Delta \vec{r}$): Difference between two position vectors. $\Delta \vec{r} = \vec{r}_2 - \vec{r}_1$

Velocity

• Average Velocity: $\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$

• Instantaneous Velocity: $\vec{v} = \frac{d\vec{r}}{dt}$ Components: $\vec{v} = v_x \hat{i} + v_y \hat{j}$

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

Acceleration

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

• Instantaneous Acceleration: $\vec{a} = \frac{d\vec{v}}{dt}$ Components: $a_x = \frac{dv_x}{dt}$, $a_y = \frac{dv_y}{dt}$

• Direction: Acceleration points toward the concave side of the path.

Constant Acceleration in 3D

• Component Equations: $v_x = v_{0x} + a_x t$ $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$ $v_y = v_{0y} + a_y t$ $y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2$ $v_z = v_{0z} + a_z t$ $z = z_0 + v_{0z} t + \frac{1}{2} a_z t^2$

• Vector Notation: $\vec{v} = \vec{v}_0 + \vec{a} t$ $\vec{r} = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2} \vec{a} t^2$

Projectile Motion

Definition and Characteristics

• Projectile Motion: The motion of an object given an initial velocity near Earth's surface, moving under gravity alone (air resistance neglected).

• Key Features:

  • Constant downward acceleration ($a_y = -g$)

  • Parabolic trajectory

Horizontal and Vertical Motion Independence

• The horizontal and vertical motions are independent.

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

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

• Projectile motion can be analyzed as a superposition of these two motions.

Projectile Motion Equations

• Initial velocity components: $v_{0x} = v_0 \cos \theta$ $v_{0y} = v_0 \sin \theta$

• Horizontal motion: $x = x_0 + v_{0x} t$

• Vertical motion: $y = y_0 + v_{0y} t - \frac{1}{2} g t^2$

Symmetric Projectile Motion

• Range ($R$): The horizontal distance traveled by the projectile. $R = v_{0x} t_{total}$

• Total time of flight (for $y_0 = 0$): $t_{total} = \frac{2 v_{0y}}{g}$

• Maximum height: $h_{max} = \frac{v_{0y}^2}{2g}$

Example: Projectile from a Cliff

• Given: Initial speed, launch angle, and height above ground.

• Find: Range and final speed using the equations above and the quadratic formula for time.

Summary Table: Key Equations for Projectile Motion

Quantity	Equation	Description
Horizontal position	$x = x_0 + v_{0x} t$	Constant velocity
Vertical position	$y = y_0 + v_{0y} t - \frac{1}{2} g t^2$	Constant acceleration
Horizontal velocity	$v_x = v_{0x}$	Unchanged (no air resistance)
Vertical velocity	$v_y = v_{0y} - g t$	Decreases linearly with time
Range (level ground)	$R = \frac{v_0^2 \sin 2\theta}{g}$	Horizontal distance for $y_0 = 0$
Time of flight (level ground)	$t = \frac{2 v_0 \sin \theta}{g}$	Total time in air for $y_0 = 0$

Conceptual Questions and Applications

• Minimum speed: At the highest point of the trajectory, the vertical velocity is zero, so the speed is minimum.

• Acceleration: The acceleration is always $-g$ downward, constant throughout the flight (ignoring air resistance).

• Independence of motion: The time to hit the ground depends only on the vertical motion, not the horizontal velocity.

• Practical example: Dropping a package from a moving airplane – the package follows a parabolic path and lands directly below if air resistance is neglected.

Summary

• 2D motion requires vector analysis for position, velocity, and acceleration.

• Projectile motion is a classic example of 2D motion with constant acceleration in the vertical direction and constant velocity in the horizontal direction.

• Key equations allow prediction of range, time of flight, and final velocity for projectiles.