Velocity and Acceleration as Derivatives of the Position Vector + Example of an Object moving in 2D

2 days ago (Edited)

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In this video, I show that the derivative of the position vector of a particle moving in 3D space is just the velocity, and likewise the acceleration is obtained as the derivative of the velocity. Thus, just knowing the position vector, we can obtain the velocity and acceleration vectors of the particle. And the speed of the object is obtained by calculating the magnitude or length of the velocity vector, which we can calculate via the distance formula. I go over an example of an object moving in 2D to illustrate this, as well as making an awesome calculator with GeoGebra!

#math #vectors #calculus #physics #Geogebra

1 Velocity and Acceleration.png

Timestamps

Motion in Space: Velocity and Acceleration – 0:00
Drawing a particle moving through 3D space – 0:57
For small values of the time increment h, the vector shown (which is the average velocity) approximates the direction of the particle along the curve – 4:41
Velocity vector is the limit of the average velocity as the time increment h approaches zero, which is the definition of the derivative of the position vector – 6:00
The speed of the particle is the magnitude of the velocity vector – 7:04
Speed is the rate of change of the distance (or arc length) with respect to time – 8:26
Acceleration is the derivative of the velocity of the particle – 9:55
Example 1: Moving Particle in 2D – 11:05
- Obtaining velocity as the first derivative and acceleration as the second derivative of the position vector – 11:48
- Speed is the length or magnitude of the velocity, and calculated using the distance formula – 13:33
- Typo: When t = 1, not t = 0 – 14:33
- Graphing the velocity and acceleration vectors – 16:33
- Going over inputs for GeoGebra 2D graphing calculator – 19:30
- Graphing in GeoGebra – 21:15
- Simplified the GeoGebra inputs – 22:40 .