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
a sin curve has a sinusoidal velocity, goes to 0 at the ends and is maximum at the center.
This has a constant speed, so square wave velocity.
I still don't quite understand this.
To determine whether the horizontal distance, denoted as d, from the center of the screen can be described by the equation d = a sin(bt), where t represents seconds, we need to understand the behavior of a sine function.
The sine function, sin(bt), is a periodic function that oscillates between -1 and 1, with the values repeating every 2π units. The variable b represents the rate at which the function oscillates, affecting the frequency of the waves. The variable a scales the amplitude, determining the maximum and minimum values of the function.
Now, let's consider the scenario in the video game with the bouncing cursor. The cursor moves horizontally across the screen at a constant rate, back and forth. Since the motion is constant, there is no oscillation or periodic behavior like that described by the sine function.
Instead, the relationship between the cursor's distance from the center of the screen (d) and time (t) is expected to be linear. As time progresses, the distance from the center increases or decreases at a constant rate. This behavior can be described by a straight-line equation, such as d = mt + c, where m is the slope and c is the y-intercept.
To further illustrate this point, let's compare the graphs of the linear equation (d = mt + c) and the sinusoidal equation (d = a sin(bt)).
Graph 1: Linear equation - d = mt + c
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In the linear graph, the distance (d) increases or decreases at a constant rate as time (t) progresses. The graph is a straight line, indicating no periodic behavior.
Graph 2: Sinusoidal equation - d = a sin(bt)
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In the sinusoidal graph, the distance (d) oscillates between maximum and minimum values over time (t). The graph forms a wave-like pattern, indicating periodic behavior.
Clearly, the linear equation aligns with the concept of a cursor moving horizontally across the screen at a constant rate, while the sinusoidal equation does not capture this behavior.
Therefore, based on the characteristics of the cursor's movement and the distinguishable nature of the sinusoidal equation, we can conclude that the horizontal distance from the center of the screen in the video game cannot be accurately modeled by the equation d = a sin(bt).