Calculus/Kinematics

 ← Centre of mass Calculus Parametric and Polar Equations → 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
• $a(t) = x^{\prime\prime}(t)$ is its acceleration function

Differentiation

Average Velocity and Acceleration

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

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

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_{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 $a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}.$
$\frac{d^2s}{dt^2} = 24t + e^t$
• Find $a(3) = \frac{d^2s}{dt^2} |_{t=3}$
 $\frac{d^2s}{dt^2} |_{t=3}$ = $24(3) + e^3 \$ = $72 + e^3 \$ = $92.08553692... \$
Answer: $a(3) = 92.08553692... \$


Integration

• $x_2 - x_1 = \int_{t_1}^{t_2} v(t) dt$
• $v_2 - v_1 = \int_{t_1}^{t_2} a(t) dt$