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
• $v(t)=x'(t)$ 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.

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

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 $[2,7]$ :

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

Answer: $v_{\rm {avg}}=50$ .

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

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

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

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

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

Integration

• $x_{2}-x_{1}=\int \limits _{t_{1}}^{t_{2}}v(t)dt$
• $v_{2}-v_{1}=\int \limits _{t_{1}}^{t_{2}}a(t)dt$