How can motion due to uniform velocity influence the distance-time graph of a particle oscillating sinusoidally?

Question

While reading "An Introduction to Mechanics" by Daniel Kleppner and Robert Kolenkow I came across the following problem statement:

"The electron is initially at rest, $x_0$ = $v_0$ = 0, so we have

x(t) = $a_0$/ ω *t − $a_0$/ $ω^2$ * sinωt.

The result is interesting: the second term oscillates and corresponds to the jiggling motion of the electron that we predicted. The first term, however, corresponds to motion with uniform velocity, so in addition to the jiggling motion the electron starts to drift away."

I know that the following equation would form a sine wave.

x(t) = − $a_0$/ $ω^2$ * sinωt

But I can't visualize how the "first term", i.e. $a_0$/ ω *t , would influence the sinusoidal graph. What type of drift would it create. If someone could explain it with a graph, I would have been very grateful.

Answer 1

Think about how the two terms act as $t$ increases. At $t=0$, both terms are zero as expected. As you begin to increase $t$, the sinusoidal term will begin to oscillate, but the first term will increase linearly. Thus, your sinusoid will shift upwards at a constant rate.

Ignoring the proportionality of the constants, a graph might look something like this: