Tangent & Normal Vectors in Motion

If a particle moves acording to r(t), unit tangent T(t) and principal unit normal N(t) tells about its movement direction. From the previous post, we know that T(t) is the unit vector of velocity, that means,  v(t) can be written in term of T(t) as

v(t) = |v(t)|T(t). 

Not only v(t) can be written in term of T(t), its derivative - acceleration a(t) can be written in terms of T(t) and N(t) as well:

a(t) = a_TT + a_NN
where a_T is named as the tangential component and a_N as normal component of acceleration. Both of these components can be found using the formulae below:

There are two formulae to find a_N as given above. Use either one that is convenient to you. 

Using the example below, you can see that the red circle is the curve for a(t). It can be obtained through adding a_TT and a_NN. Slide the value of t if you want to move point C on the circle. If you untick the curve a(t) and tick for v(t), you will find a curve for v(t) and the curve is obtained through the vector  |v(t)|T(t) from the origin to point B. Slide the value of t if you want to move point B on the cirlce. Untick v(t) and tick for r(t) now, this curve is the curve of position vector r(t). You can see that the unit tangent T(t) and its principal unit normal N(t) at point A. 

If you tick all curves, you will find that all vectors of T(t), |v(t)|T(t) and a_TT are parallel, only differs in length. Same goes to N(t) and a_NN. Any other interesting stuff you can find from this graph? If you want to rotate the graph, just click here.

Betty, created with GeoGebra and Thomas' Calculus, Pearson.