Calculus/Kinematics

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Kinematics

Contents

[edit] 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:

[edit] Differentiation

[edit] 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}

[edit] 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 \ .

[edit] 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}

[edit] 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... \

[edit] 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

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