I already asked this, but I didn't really understand the answer I got. I got the general idea, but I don't have a good enough understanding to answer the question.
Writing to Learn: In a certain video game a cursor bounces back and forth horizontally across the screen at a constant rate. Its distance d from the center of the screen varies with time t and hence can be described as a function of t. Explain why this horizontal distance d from the center of the screen does not vary according to an equation d= a sin bt, where t represents seconds. You may find it helpful to include a graph in your explanation
To understand why the horizontal distance, d, from the center of the screen in the video game does not vary according to the equation d = a sin(bt), let's break down the components of the equation.
The equation d = a sin(bt) represents a sinusoidal function, where a is the amplitude (maximum distance from the center), b is the angular frequency (rate at which the cursor oscillates), and t represents time in seconds.
In the given scenario, the cursor bounces back and forth horizontally across the screen at a constant rate. This indicates that the cursor's motion is linear rather than sinusoidal. In other words, the cursor's distance from the center changes linearly rather than in a sinusoidal pattern.
To illustrate this, let's compare a sinusoidal function and a linear function on a graph. Imagine the x-axis represents time (t), and the y-axis represents the distance of the cursor from the center (d).
A sinusoidal function would produce a wave-like pattern on the graph, where the distance oscillates smoothly between positive and negative values. On the other hand, a linear function would result in a straight line graph, indicating a constant rate of change.
If the cursor's motion followed the sinusoidal equation d = a sin(bt), the graph would show wave-like oscillations. However, because the cursor's motion is linear, the graph would instead be a straight line.
Therefore, the equation d = a sin(bt) does not accurately describe the horizontal distance of the cursor from the center in this particular video game scenario. Instead, a linear equation or a different function should be used to represent the cursor's motion accurately.