1.2: Velocity Vectors and Speed

     Parametrizing a curve by t suggests considering the curve as the locus traced out by a point in motion. If the coordinate functions of the curve are differentiable, then at each point we have a well-defined velocity vector with coordinate functions x'(t), y'(t).


This demo allows you to trace out the position and velocity vectors of a curve X(t). There is a tapedeck that controls the value of a variable t0. As you step through the values of t0, a position vector and a velocity vector are drawn for each value, in red and blue, respectively.



1. For a circle, X(t) = (cos(t),sin(t)) with 0 ≤ t < 2π, how does the direction of the velocity vector relate to the direction of the position vector? What happens if the circle is deformed to an ellipse?

2. Consider the exponential spiral X(t) = ect(cos(t), sin(t)). How does the direction of the velocity vector relate to the direction of the position vector? Explain what happens as c changes in general and what happens when c equals zero.

3. How does the direction of the velocity vector relate to the direction of the position vector for the arithmetic spiral, X(t) = t(cos(t), sin(t))?

4. What happens to the velocity vector at the origin for the curve X(t)=(t3,t4)? Why?

5. Consider the general polar coordinate curve, X(t) = (r(t)cos(t), r(t)sin(t)). Under what circumstances will the velocity vector and the position vector be perpendicular? When will they be linearly independent?
The speed of a parametrized curve at a point X(t) is defined to be the length of the velocity vector at the point. We can obtain this length by taking the square root of the dot product of the velocity vector with itself,

√(X'(t)·X'(t))

We denote the speed function by either s'(t) or ds/dt. The speed may be thought of as the rate of change of the length of the curve with respect to time. We can then integrate the speed from t=a to t=b, the endpoints of the domain, to get the total length of the curve between the endpoints X(a) and X(b).

s(b)-s(a) = ∫ab√(X'(t)·X'(t))dt = ∫ab|X'(t)|dt

More generally, the arclength function s(t) of a curve from a point s(t0) to a variable point s(t) is given by

s(t)-s(t0) = ∫t0t√|X'(u)|du

Arclength is one of the few properties that a curve has. It is a "natural parameter", and for a curve without singularities, it provides a way of "uniformizing" a curve. If you make a trip along a road, you can note the time at which you pass various landmarks, and then someone else could follow the same route and know when to look up to see a particular sight. But in order to do that, the second person would also have to know the velocity at each time. More uniform is the standard process of indicating noteworthy sights by telling how far along the route they are, for example, a rest stop at milestone 134 on the Ohio Turnpike, or a castle twenty kilometers south of Bonn on the road along the Rhein. Of course it is possible to use any unit of measurement, miles, kilometers, or what have you, but two travelers using the same units can be sure to be at the same place at the same time if they walk at one unit per hour.


This demo shows a plane curve X(u), the graph of the speed s'(u) over the domain, and the graph of the distance s(u) along the curve over the domain. The tapedeck controls the value of t. In the Speed versus t window, a cyan curve draws the speed s'(t), and an area is swept out underneath it as it moves across the graph. This area is equal to the distance s(t) traveled along the curve, shown in the "Distance" window.



1. Describe the distance functions and the speed functions for a function graph of the form X(t) = (t,tm) for various values of m.

2. What does the distance function look like for a curve with a cusp, for example, X(t) = (t4,t5)? When will the distance function have a horizontal tangent?

3. Calculate s(t) explicitly for a circle and for a straight line. Enter the equations for a circle and for a straight line in the demo (for example, the circle (cos(t),sin(t)), and the line (t,2t)) and compare your expressions for the arclength with the graphs in the demo. More generally you can look at a circle of radius R and a line with slope m.

4. Describe the functions s(t) and s'(t) for the exponential spiral, X(t) = et(cos(t,sin(t))).
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