Example 3: Integrating Acceleration and Velocity Vectors to get the Position Vector

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In this video, I show that we can obtain the velocity of a moving object by first integrating its acceleration vector to obtain its velocity and then integrating the velocity vector. This follows from my earlier videos on derivatives and integrals of vector functions by their components. I illustrate this with an example and show that the integration constants can be solved by plugging in the initial time variable, in our case t = 0. Thus, the velocity vector is the initial velocity plus the definite integral of the acceleration, while the position vector is the initial position plus the definite integral of the velocity. I also graph out the velocity and acceleration vectors along the position curve with the GeoGebra 3D graphing calculator: https://www.geogebra.org/3d/pbrywwev

#math #vectors #calculus #physics #Geogebra

Timestamps

• Recap on derivatives and integrals of vector functions – 0:00
• Example 3: Integrating to get velocity and position vectors – 1:37
• Solution: Integrating acceleration to get velocity – 2:24
  • Plug in t = 0 to determine the integration constant vector – 4:57
  • Integrating velocity to get position – 7:41
  • Plug in t = 0 to determine the integration constant vector – 10:33
  • Graph of the position vector at t = 3 – 12:28
  • Inputs for the GeoGebra 3D graphing calculator – 15:37
  • Graphing in GeoGebra – 17:49
  • Summary: Vector integrals recover velocity when acceleration is known and position when velocity is known – 19:00 .

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