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

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 $$[2,7]$$ :




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

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


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


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




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

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

Integration

 * $$x_2-x_1=\int\limits_{t_1}^{t_2} v(t)dt$$
 * $$v_2-v_1=\int\limits_{t_1}^{t_2} a(t)dt$$