Plane Curvilinear Motion - n-t Coordinates

Notice that the acceleration equation (above) includes taking the derivative of the unit direction vector e_t. Does de_t /dt = 0 like di/dt = 0?

The following information will show how the following n-t acceleration equation is derived.

a = (dv/dt)e_t + (v²/ρ)e_n

Which equation represents the time derivative of the velocity that will give an expression for the acceleration.

v = v e_t

This is not a very useful form of the acceleration equation. We need to find out what de_t /dt equals. We need to know both the magnitude and direction of de_t /dt. If we look at the above figure, we see that as ds gets smaller and smaller de_t begins to point in n-direction.

Direction: de_t = de_t e_n

The magnitude of de_t can be determined by using the arc length equation. Looking at the figure and noting that the magnitude of a unit direction vector is equal to one, we get the following.

Magnitude: |de_t| = |e_t| dθ = 1 dθ

Taking the time derivative of the magnitude equation we get

de_t /dt = dθ/dt

Recalling that the velocity equals

v = ρ(dθ/dt)

Substituting the velocity equation and the magnitude and direction information into the acceleration equation we obtain a useful version of the acceleration equation.

a = (dv/dt)e_t + v (de_t /dt) = (dv/dt)e_t + (v²/ρ)e_n

A particle’s acceleration is always nonzero if it is moving on a curve, even if its speed is constant. Why?

What does (dv/dt) physically represent in the acceleration equation? a = (dv/dt)e_t + (v²/ρ)e_n

What does (v²/ρ) physically represent in the acceleration equation? a = (dv/dt)e_t + (v²/ρ)e_n