Calculus/Kinematics

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Kinematics

Introduction[edit]

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:

Differentiation[edit]

Average Velocity and Acceleration[edit]

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[edit]

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[edit]

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

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

Examples[edit]

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
Answer: a(3)=92.08553692\dots

Integration[edit]

  • 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