A function can be thought of as a curve in . Write the curve in parametric form

Think of the parameter t as *time* and the curve as being
traced out by a moving object, so that the object is at the position
at time t.

With this interpretation:

(a) is the * velocity
vector* of the object. It points in the direction that the object
is travelling at time t. Its length is the *
speed* of the object at time t.

(b) is the * acceleration
vector* of the object. It represents the direction and magnitude
of the change of the velocity vector at time t.

* Example.* Find the velocity and acceleration
vectors at for the curve with position
function

The velocity is the derivative of the position:

The acceleration is the derivative of the velocity:

Setting gives

* Example.* The position of an evil lime jello at
time t is

Find its velocity vector and its speed at .

The velocity is the derivative of the position:

The velocity at is

The speed is the length of the velocity vector:

* Example.* A cheeseburger moves on a * circular helix* given by

Show that it moves with constant speed.

The velocity vector is

The speed is the length of the velocity vector, which is

Thus, the cheeseburger moves with constant speed.

* Example.* Prove that if a curve has constant length, then its velocity and position
vectors are always perpendicular.

I'll use the fact that the square of the length of a vector equals the dot product of the vector with itself:

Suppose , where r is a constant. Then use the identity above, differentiate, and apply the Product Rule for dot products:

Since and have dot product 0, they are perpendicular.

Note: To say that has constant length r means that a point on the curve stays a constant distance r from the origin. Thus, it must be moving on the sphere of radius r centered at the origin.

Since , you can integrate to find the position function from the velocity:

Likewise, since , you can integrate to find the velocity from the acceleration:

Antiderivatives are only determined up to an arbitrary constant. But
you may be able to determine the arbitrary constant if you are given
* initial conditions*.

* Example.* The acceleration vector for a bacon
quiche is

Find the position function , if and .

Now , so

Hence,

Next,

Now , so

Hence,

Copyright 2018 by Bruce Ikenaga