Linear Motion: Position, Velocity, Acceleration, and Integration in Physics

Linear Motion: Position, Velocity & Acceleration

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. The primary quantities in kinematics are position, velocity, and acceleration, which are functions of time and can be analyzed using graphs and equations.

• Position (x): The location of a particle in space at a given time.

• Velocity (v): The rate of change of position with respect to time.

• Acceleration (a): The rate of change of velocity with respect to time.

Equations of Motion

For a particle with motion described by $x = 6t^2 - t^3$, the following derivatives apply:

• Velocity: $v = \frac{dx}{dt} = 12t - 3t^2$

• Acceleration: $a = \frac{dv}{dt} = 12 - 6t$

These equations allow us to determine the particle's position, velocity, and acceleration at any time t.

Example Calculations

• At $t = 0$: $x = 0$, $v = 0$, $a = 12$ m/s2

• At $t = 2$ s: $x = 16$ m, $v = 12$ m/s, $a = 0$

• At $t = 4$ s: $x = 32$ m, $v = 0$, $a = -12$ m/s2

Motion Diagrams

Understanding Motion Diagrams

A motion diagram is a visual tool that represents the position of an object at successive time intervals. It helps bridge the gap between a verbal description and a physicist's quantitative analysis.

• Each dot represents the object's position at a specific time.

• Arrows indicate velocity and acceleration vectors.

Application: Sprinter's Motion

Strobe photos can be used to create motion diagrams for moving objects, such as a sprinter. By measuring the distance between images and dividing by the time interval, the average speed can be calculated.

Position vs. Time Graphs

Graphical Analysis

Position vs. time graphs are essential for visualizing how an object's position changes over time. The slope of the graph at any point gives the instantaneous velocity.

• Displacement ($\Delta x$): The change in position between two times.

• Distance Traveled: The total length of the path taken, which may differ from displacement if the path is not straight.

Example

• Between $t = 0$ and $t = 17$ s: Displacement: $\Delta x = x_2 - x_1 = 28$ m Distance Traveled: $d = 30$ m + $2$m = $32$ m

Average Speed and Velocity

Definitions

• Average Speed ($v_{\text{avg}}$): Total distance traveled divided by total time taken. $v_{\text{avg}} = \frac{\text{distance}}{\text{time}}$

• Average Velocity ($\bar{v}$): Displacement divided by time interval. $\bar{v} = \frac{\Delta x}{\Delta t}$

Example Calculation

• For $d = 100$ m in $9.81$ s: $v_{\text{avg}} = \frac{100}{9.81} = 10.16$ m/s

Instantaneous Velocity and Speed

Instantaneous Velocity

The instantaneous velocity at time t is the slope of the tangent to the position vs. time graph at that point.

• $v = \frac{dx}{dt}$

• Graphically, it is the gradient of the position-time curve at a specific instant.

Instantaneous Speed

Instantaneous speed is the magnitude of instantaneous velocity and is always positive.

• $v_{\text{inst}} = \left| \frac{dx}{dt} \right|$

Acceleration

Instantaneous Acceleration

Acceleration is the rate of change of velocity with respect to time. On a velocity vs. time graph, the slope at any point gives the instantaneous acceleration.

• $a = \frac{dv}{dt}$

• Example: If $v$ changes from $4$m/s to $0$ m/s over $4$s,$a = \frac{0 - 4}{12 - 8} = -1$ m/s2

Integration in Kinematics

Indefinite and Definite Integrals

Integration is a mathematical tool used to find areas under curves, which in physics often represent quantities like displacement or velocity.

• Indefinite Integral: $\int f(x) dx$ gives a family of functions plus a constant.

• Definite Integral: $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.

Basic Table of Integration

The following table summarizes common indefinite integrals:

Function	Indefinite Integral
$x^n$	$\frac{x^{n+1}}{n+1} + c$
$e^{ax}$	$\frac{1}{a}e^{ax} + c$
$\sin(ax)$	$-\frac{1}{a}\cos(ax) + c$
$\cos(ax)$	$\frac{1}{a}\sin(ax) + c$
$\frac{1}{x}$	$\ln|x| + c$

Definite Integrals and the Fundamental Theorem of Calculus

• To evaluate $\int_a^b f(x) dx$, find the antiderivative $F(x)$, then compute $F(b) - F(a)$.

• Example: $\int_0^2 x^2 dx = \left. \frac{x^3}{3} \right|_0^2 = \frac{8}{3} - 0 = \frac{8}{3}$

Summary Table: Kinematic Quantities

Quantity	Definition	Equation
Displacement	Change in position	$\Delta x = x_2 - x_1$
Distance	Total path length	Sum of all segments
Average Speed	Distance / time	$v_{\text{avg}} = \frac{d}{t}$
Average Velocity	Displacement / time	$\bar{v} = \frac{\Delta x}{\Delta t}$
Instantaneous Velocity	Slope of position-time graph	$v = \frac{dx}{dt}$
Instantaneous Acceleration	Slope of velocity-time graph	$a = \frac{dv}{dt}$

Applications and Examples

• Analyzing the motion of a sprinter using motion diagrams and position-time graphs.

• Calculating average and instantaneous velocities from experimental data.

• Using integration to determine displacement from velocity-time graphs.

Additional info: Integration is a foundational tool in physics for connecting rates of change (derivatives) to accumulated quantities (areas under curves), especially in kinematics and dynamics.