May 07, 2026

Tangential and Normal Components of Acceleration as Derivatives of Position Vector + Example

30 minutes

https://3speak.tv/watch?v=mes/e0b98544
In this video, I rewrite our earlier equations for the tangential and normal components of acceleration to be in terms of the position vector and its derivatives. This simplifies and standardizes the components of acceleration and involves the dot product for the tangential acceleration and the cross product for the normal acceleration. Interestingly, the acceleration vector always lies on the osculating plane because it has no binormal component. The normal acceleration increases by a factor of velocity squared so that doubling the velocity multiplies the normal acceleration by 4 when along a curved path. I also go over an example of a moving particle to show how to obtain the components of acceleration in a straightforward way.

#math #vectors #calculus #physics #education

Timestamps:

Acceleration vector as tangential and normal components – 0:00

Acceleration vector is always in the osculating plane (contains both tangent and normal vectors) – 1:00

Normal component experiences greater acceleration in curved paths than does the tangential component – 3:29

Take dot product of velocity and acceleration to try to get acceleration in terms of position vector and its derivatives – 6:36

Use geometric equation of dot product to simplify results – 10:00

Tangent component of acceleration in terms of derivatives of position vector – 14:00

Normal component of acceleration in terms of derivatives of position vector (via curvature formula) – 15:50

Example 7: Tangential and Normal Components of Acceleration of a moving particle – 18:54

Solution: Get first and second derivatives of the position vector – 19:26

Tangential acceleration component via dot product – 21:37

Normal acceleration component via cross product – 23:33

Solve magnitude of cross product to solve for normal acceleration component – 27:56

Simplify terms to get final result – 29:15

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