Simple Motion Made Complex

Calculus Level 3

As an object moves in a plane let its position be represented by the complex function

$\large z(t) = A \, e^{i \omega t}$

where A is a positive number, $\omega$ is a real number, and $t$ is a real variable representing time.

What type of motion does the object have?

Note : There are six choices. If necessary scroll down to see them.

Hint : Here the $xy$ -plane has been replaced with the complex plane. $z$ is NOT the Cartesian co-ordinate. This is motion within a plane. The number $z$ can be thought of as the displacement of the object from the number zero.

The object moves along a path shaped like

y = cos(\omega t)

and has constant speed A straight line and has speed

v = \omega ( \: sin(\omega t) - cos(\omega t) \: )

The object moves in a straight line and has a constant speed. The object moves in a circle and has speed

v = \omega \: cos( \omega t)

The object moves in a circle and has a constant speed A path shaped like

y = cos(\omega t)

and has speed

v = \omega \: sin( \omega t)

1 solution

Brandon Stocks
Jun 3, 2016

The absolute value of z is the distance of the object from the number zero. Note that $e^{i \theta }$ has an absolute value of one for all real numbers $\theta$ .

$|z| = A \, |e^{i \omega t}| = A$

The object's distance from the number 0 is constant. This eliminates all choices except for the two with circular motion. The object has some kind of circular motion around the number 0, and that circle has a radius of A.

Velocity, v, is the time-derivative of position.

$v = \frac{dz}{dt} = i \omega A \, e^{i \omega t}$

$speed = |v| = A \, |\omega | \, |i| \, |e^{i \omega t}| = A \, |\omega |$

$\omega A$ is a constant, so the object is moving with a constant speed.

$\omega$ is called the angular velocity. Here it is in units of radians per unit time. If $\omega$ is positive the object is revolving counter-clockwise around 0. If $\omega$ is negative the object is revolving clockwise around 0. Students of physics may want to look at the second derivative of z. Finding the second derivative is another way of finding the equation for centripetal acceleration.

Simple Motion Made Complex

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