Linear Motion with Constant Acceleration & Free Fall in 1D (PHY 121 - University Physics I)

Linear Motion with Constant Acceleration

Introduction to 1D Motion

Linear motion in one dimension (1D) is a foundational concept in physics, describing how objects move along a straight line. This topic covers the relationships between position, velocity, and acceleration, especially under conditions of constant acceleration.

• Position: The location of an object along a coordinate axis, typically denoted as x or y.

• Displacement: The change in position, $riangle x = x_f - x_i$.

• Velocity: The rate of change of position. Average velocity is $ar{v}_x = rac{ riangle x}{ riangle t}$, and instantaneous velocity is $v_x = rac{dx}{dt}$.

• Acceleration: The rate of change of velocity. Average acceleration is $ar{a}_x = rac{ riangle v_x}{ riangle t}$, and instantaneous acceleration is $a_x = rac{dv_x}{dt}$.

Table: Quantities of 1D Motion

Quantity	Definition	Units
Position	Coordinate on an axis (x-coordinate)	m
Displacement (vector)	$riangle x = x_f - x_i$	m
Average Velocity (vector)	$v_{x,avg} = rac{ riangle x}{ riangle t}$	m/s
Instantaneous Velocity (vector)	$v_x = rac{dx}{dt}$	m/s
Average Acceleration (vector)	$a_{x,avg} = rac{ riangle v_x}{ riangle t}$	m/s2
Instantaneous Acceleration (vector)	$a_x = rac{dv_x}{dt}$	m/s2

Speeding Up and Slowing Down

The relationship between velocity and acceleration determines whether an object speeds up or slows down:

• Speeding Up: If velocity and acceleration are in the same direction, the object speeds up.

• Slowing Down: If velocity and acceleration are in opposite directions, the object slows down.

Equations of Motion

Motion with Constant Velocity

When velocity is constant, the position changes linearly with time:

• Equation: $x(t) = x_0 + v_x (t - t_0)$

• Example: A car moving at a steady speed on a straight road.

Motion with Constant Acceleration

When acceleration is constant, the following equations describe the motion:

• Velocity as a function of time: $v_x(t) = v_{0x} + a_x t$

• Position as a function of time: $x(t) = x_0 + v_{0x} t + rac{1}{2} a_x t^2$

• Velocity-position relationship (eliminating time): $v_x^2 = v_{0x}^2 + 2 a_x riangle x$

Integration Approach (General Case)

If acceleration or velocity is given as a function of time, integration is used:

• $v_x(t_f) - v_x(t_i) = extstyleigint_{t_i}^{t_f} a_x(t) dt$

• $x(t_f) - x(t_i) = extstyleigint_{t_i}^{t_f} v_x(t) dt$

Worked Example: Constant Acceleration

Given: $v_{0x} = 20$ m/s, $a_x = -3$ m/s2, $t = 8$ s. Find $v_x$ after 8 s:

• $v_x = v_{0x} + a_x t$

• $v_x = 20 + (-3)(8) = -4$ m/s

Worked Example: Police Car Chase

A motorcyclist moves at 20 m/s, a police car starts from rest with $a_x = 4$ m/s2. Find time to catch up:

• Motorcyclist: $x_1 = 20t$

• Police car: $x_2 = rac{1}{2} (4) t^2 = 2t^2$

• Set $x_1 = x_2$: $20t = 2t^2 ightarrow t = 10$ s

Free Fall Motion in 1D

Introduction to Free Fall

Free fall describes motion under the influence of gravity alone. Near Earth's surface, all objects experience a constant downward acceleration, $g = 9.8$ m/s2.

• Gravitational acceleration: $a_y = -g$

• Equations for vertical motion:

  • $v_y = v_{0y} + a_y t$

  • $y = y_0 + v_{0y} t + rac{1}{2} a_y t^2$

  • $v_y^2 = v_{0y}^2 + 2 a_y riangle y$

Example: Ball Thrown Upward

For a ball thrown upward with initial speed $v_{0y}$ from height $y_0$:

• Maximum height can be found using $v_y^2 = v_{0y}^2 + 2 a_y riangle y$ (set $v_y = 0$ at the top).

Graphical Analysis

Velocity vs. time graphs for free fall show linear changes due to constant acceleration. The direction and slope indicate whether the object is moving upward or downward and whether it is speeding up or slowing down.

Summary Table: Key Equations for 1D Motion

Scenario	Equation
Constant Velocity	$x(t) = x_0 + v_x t$
Constant Acceleration (velocity)	$v_x(t) = v_{0x} + a_x t$
Constant Acceleration (position)	$x(t) = x_0 + v_{0x} t + rac{1}{2} a_x t^2$
Constant Acceleration (velocity-position)	$v_x^2 = v_{0x}^2 + 2 a_x riangle x$
Free Fall (vertical)	$a_y = -g$

Additional info:

• These notes are based on lecture slides for PHY 121, University Physics I - Mechanics, focusing on Chapter 2: Linear Motion with Constant Acceleration and Free Fall in 1D.

• Next topics include motion in two dimensions and projectile motion.