Calculus/Kinematics

Introduction

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

If $x$ is the position of some moving object, and $t$ is time, this section uses the following conventions:

• $x(t)$ is its position function
• $v(t)=x^{\prime }(t)$ is its velocity function
• $a(t)=x^{″}(t)$ is its acceleration function

Differentiation

Average Velocity and Acceleration

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

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

Examples

Example 1:

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

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

$v_{\mathrm{a}\mathrm{v}\mathrm{g}}$	$=\frac{x(7)-x(2)}{7-2}$
	$=\frac{252-2}{5}$
	$=50$

Answer: $v_{\mathrm{a}\mathrm{v}\mathrm{g}}=50$ .

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

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

Answer: $a(3)=92.08553692\dots$

Integration

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