equation of motion of roller coaster that y=sin(t/2),y=(t-1)(t-2)(t-3)(t-4)(t-5)
To find the equation of motion for the roller coaster, we need to combine the given equations: y = sin(t/2) and y = (t-1)(t-2)(t-3)(t-4)(t-5).
Let's break it down step by step:
First, let's focus on the equation y = sin(t/2). This represents the vertical displacement of the roller coaster at any given time t. The t/2 within the sine function is responsible for controlling the speed of oscillation of the coaster. As t increases, the value within the sine function increases, resulting in a faster oscillation. The sine function values range from -1 to 1, which will determine the amplitude (maximum vertical displacement) of the coaster.
Now, let's consider the second equation y = (t-1)(t-2)(t-3)(t-4)(t-5). This equation represents a polynomial function of degree 5, which will determine the shape and position of the roller coaster. The factors (t-1), (t-2), (t-3), (t-4), and (t-5) represent the roots of the polynomial function. When t = 1, 2, 3, 4, or 5, each factor becomes zero, resulting in vertical displacement y = 0 at those time points. These roots will determine turning points or critical points of the coaster's path.
To obtain the combined equation of motion, we need to consider both equations simultaneously. This means that the roller coaster will follow both the sine curve and the polynomial curve simultaneously.
The combined equation of motion for the roller coaster is: y = (t-1)(t-2)(t-3)(t-4)(t-5) + sin(t/2)
This equation represents both the shape and position of the coaster as it oscillates vertically and moves horizontally along the x-axis.
Keep in mind that this equation assumes a vertical displacement (y-axis) and can be plotted against the time (t) on the horizontal axis.