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_{avg} = \frac{\Delta x}{\Delta t}
  •  a_{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_{avg} \ =  \frac{x(7) - x(2)}{7-2}
=  \frac{252 - 2}{5}
=  50 \
Answer:  v_{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} |_{t=3}
 \frac{d^2s}{dt^2} |_{t=3} =  24(3) + e^3 \
=  72 + e^3 \
=  92.08553692... \
Answer:  a(3) = 92.08553692... \ 

Integration[edit]

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