DText 3.2

3.2: Velocity Vectors and Speed

If we consider the parameter t to represent time, several questions arise. If you were travelling along a curve X(t) in a car whose position, how would you feel your position changing in time? What forces would you feel when you reached some point where the curve took an interesting twist? We can answer these questions by looking at the time derivatives of the position vector. Because we are using vector notation rather than coordinate notation, we can use the definition for the derivative of a vector function in the plane to give the derivative of a vector function in three-space as well. We define the first derivative of the function X(t) to be the following limit, assuming that the limit exists:

X'(t) = lim (h → 0) (X(t+h)-X(t))/h
When we think of the parameter as representing time, then the derivative X'(t) of the position vector is sometimes called the velocity vector, and is denoted V(t).

This demo illustrates what happens to the difference quotient V(t,h) as h tends to 0 . You can scroll through the values of h. In pink are the velocity vectors, and in yellow are the vectors V(t,h) = (X(t + h) - X(t))/h.

The goal in using this demo is to familiarize oneself with the velocity vectors of a space curve and how they relate to the position vector. The velocity vectors are traced out in pink, while the position vectors are traced out in orange, as you scroll through the tapedeck for t0.

1. Consider the relationship of the velocity vector to the position vector for various curves - for example, a helix X(t) = (c*cos(t),c*sin(t),b*t) or a conical spiral X(t) = (e^tcos(t),e^tsin(t),t).

Speed and Arclength in Three-space

Arclength in three dimensions is a natural extension of arclength in two dimensions. As in two dimensions, the speed of a parameterized curve at a point t is defined to be the length of the velocity vector at the point, which is √(X'(t) · X'(t)). As before, we designate the speed by s'(T) or ds/dt, indicating the ratio of change of distance along the curve with respect to time. We can then integrate the speed from t=a to t=b to get the total length of the curve:

s(b) - s(a) = ∫_a^b√(X'(t) · X'(t))dt.
More generally, the arclength of a curve between t₀ and t is given by:

s(t) - s(t₀) = ∫_t₀^t√(X'(u) · X'(u))du.

In this demo you can enter a curve in the control window and see three graphs: the graph of the curve, the graph of s'(t) - the velocity curve, and the graph of s(t) - the distance along the curve versus the variable t . If the t0 tapedeck is moving forward, a point will move along the velocity curve, leaving vertical lines as it goes. This illustrates the notion that the distance along the curve is equal to the area under between the function graph of the speed and the t-axis.

1. What can we say about the speed and arclength of a circular helix? (You can calculate s(t) for yourself and compare your results with the graph in the demo.) At which points would a car on the curve be moving fastest?

2. What can we say about the speed and arclength of an exponential spiral on a cone? At which points would a car on the curve be moving fastest?

3. Same question for an ellipse, or for a space cardioid. What about the twisted cubic?

4. What happens on a curve if the distance function has a horizontal tangent? Find an example of such a curve.