Kinematics: Motion Along a Straight Line and Free Fall

Chapter 2: Motion Along a Straight Line

Introduction

This chapter covers the fundamental concepts of kinematics, focusing on motion along a straight line. Key topics include displacement, velocity, acceleration, and the equations of motion for objects experiencing constant acceleration.

Displacement and Position

• Displacement is the change in position of an object, defined as $\Delta x = x_2 - x_1$.

• Position is the location of an object at a particular time, usually measured along a straight line (x-axis).

• Example: If a car moves from $x_1 = 0$ m to $x_2 = 50$ m, its displacement is $50$ m.

Average Velocity and Instantaneous Velocity

• Average velocity is the total displacement divided by the total time interval: $v_{x,avg} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1}$

• Units: m/s

• Instantaneous velocity is the velocity at a specific moment, defined as: $v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

• Graphical interpretation: The slope of the position vs. time graph gives velocity.

• Example: If the position-time graph is linear, the velocity is constant; if curved, velocity changes.

Acceleration

• Acceleration is the rate of change of velocity with respect to time: $a_x = \frac{\Delta v_x}{\Delta t}$

• Units: m/s2

• Speeding up: $v_{2x} - v_{1x} > 0 \Rightarrow +a_x$

• Slowing down: $v_{2x} - v_{1x} < 0 \Rightarrow -a_x$

• Example: A car increasing its speed from 10 m/s to 20 m/s in 5 seconds has $a_x = 2$ m/s2.

Equations of Motion for Constant Acceleration

• 1. Acceleration formula: $a_x = \frac{v_{2x} - v_{1x}}{t}$

• 2. Position as a function of time: $x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$

• 3. Velocity as a function of position: $v_x^2 = v_{0x}^2 + 2 a_x \Delta x$

• 4. Position/velocity/time relationships: Use the above equations to solve for any variable when the others are known.

• Example: An object starts at $x_0 = 0$ m with $v_{0x} = 5$ m/s and $a_x = 2$ m/s2. After $t = 3$ s, $x = 0 + 5 \times 3 + \frac{1}{2} \times 2 \times 9 = 15 + 9 = 24$ m.

Worked Example: Train Motion

• A train accelerates at $0.5$ m/s2 for $60$seconds, maintains constant speed for $240$ seconds, and slows down at $-1.0$ m/s2 for $30$ seconds.

• Step 1: Calculate displacement during acceleration: $x = \frac{1}{2} a t^2 = \frac{1}{2} (0.5) (60^2) = 900$ m$

• Step 2: Final velocity after acceleration: $v_x = \sqrt{2 a_x \Delta x} = \sqrt{2 \times 0.5 \times 900} = 30$ m/s$

• Step 3: Displacement during constant speed: $x = v t = 30 \times 240 = 7200$ m$

• Step 4: Displacement during deceleration: Use $x = v_0 t + \frac{1}{2} a t^2$ with $a = -1.0$ m/s2, $v_0 = 30$ m/s, $t = 30$ s.

Free Fall

Introduction

Free fall describes motion where gravity is the only force acting on an object. The acceleration due to gravity varies depending on the celestial body.

Celestial Body	Acceleration due to Gravity (m/s2)
Earth	9.81
Moon	1.62
Sun	274

Equations for Free Fall (Gravity)

• Vertical velocity: $v_y = v_{0y} - g t$

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

• Note: Never combine x and y equations; treat each direction separately.

• Example: An object dropped from rest ($v_{0y} = 0$) falls for $t = 2$ s. Its velocity is $v_y = -9.81 \times 2 = -19.62$ m/s, and its position is $y = 0 - \frac{1}{2} \times 9.81 \times 4 = -19.62$ m.

Summary Table: Kinematic Equations

Equation	Physical Meaning
$v_x = v_{0x} + a_x t$	Velocity as a function of time
$x = x_0 + v_{0x} t + \frac{1}{2} a_x t^2$	Position as a function of time
$v_x^2 = v_{0x}^2 + 2 a_x \Delta x$	Velocity as a function of position
$a_x = \frac{v_{2x} - v_{1x}}{t}$	Definition of acceleration

Additional info:

• These notes are suitable for introductory college physics courses covering kinematics and free fall.

• Graphs and diagrams referenced in the notes illustrate position-time and velocity-time relationships, which are essential for understanding motion.