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In this video, I write the vector equation of acceleration in terms of its unit tangent and unit normal vector components, which is useful when analyzing the motion of a particle along a curve. The unit normal vector is defined as the ratio of the tangent vector (derivative of the position vector) with its magnitude; hence, the length will be 1. Likewise, the unit normal vector is the derivative of the unit tangent vector divided by its magnitude. I use the equation for the curvature and derivative of velocity to obtain the acceleration in terms of the unit tangent and unit normal vectors. These components outline how much acceleration an object experiences in the direction of its current motion (tangent) and how much in the direction perpendicular to its motion (normal). This is useful in calculating the forces associated with such accelerations.
#math #vectors #calculus #physics #education
Timestamps
Tangential and Normal Components of Acceleration – 0:00
Recall the unit tangent, unit normal, and unit binormal vectors – 0:38
Velocity in terms of the unit tangent vector T – 2:10
Derivative of velocity obtains the acceleration vector in terms of T and T’ – 4:26
Recap on arc length and curvature – 5:31
T’ in terms of curvature – 7:32
Unit normal tangent vector is defined in terms of T’ – 9:11
Acceleration in terms of tangential and normal components – 11:19
Illustration of the components of acceleration – 13:44
Length of the unit vectors is 1 – 16:33
