Calculus/Kinematics

Introduction

Kinematics or the study of motion is a very relevant topic in calculus.

This section uses the following conventions:

• $x(t)$ represents the position equation
• $v(t)$ represents the velocity equation
• $a(t)$ represents the acceleration equation

Differentiation

Average Velocity and Acceleration

Average velocity and acceleration problems use the algebraic definitions of velocity and acceleration.

• $v_{avg}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}$
• $a_{avg}=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$

Examples

Example 1:

A particle's position is defined by the equation $x(t)=x^3-2x^2+x$. Find the
average velocity over the interval [2,7].

• Find the average velocity over the interval [2,7]:

$v_{avg}$	=	$\frac{x(7)-x(2)}{7-2}$
	=	$\frac{252-2}{5}$
	=	$50$

Answer: $v_{avg}=50$.

Instantaneous Velocity and Acceleration

Instantaneous velocity and acceleration problems use the derivative definitions of velocity and acceleration.

• $v(t)=\frac{dx}{dt}$
• $a(t)=\frac{dv}{dt}$

Examples

Example 2:

A particle moves along a path with a position that can be determined by the function $x(t)=4t^3+e^t$. 
Determine the acceleration when $t=3$.

• Find $v(t)=\frac{ds}{dt}.$

$\frac{ds}{dt}=12t^2+e^t$

• Find $a(t)=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}.$

$\frac{d^{2}s}{d{t}^2}=24t+e^t$

• Find $a(3)=\frac{d^{2}s}{d{t}^2}{|}_{t=3}$

$\frac{d^{2}s}{d{t}^2}{|}_{t=3}$	=	$24(3)+e^3$
	=	$72+e^3$
	=	$92.08553692...$

Answer: $a(3)=92.08553692...$

Integration

• $x_2-x_1=\underset{t_1}{\overset{t_2}{\int }}v(t)dt$
• $v_2-v_1=\underset{t_1}{\overset{t_2}{\int }}a(t)dt$