A weight attached to a spring is pulled down so that it is 10 cm from the floor and is released so that it bounces up and down. When the effects of friction and gravity are ignored. It's height can be modeled by a sine function of the time since it started bouncing. The weight reaches its first maximum height of 50 cm at 1.5 s.
a. Write an equation for the height, in cm,of the weight as a function of time, in seconds.
b. graph the equation from part a.
c. When is the weight moving up fastest?
d. When is it moving down faster?
e. at what time is the weight changing direction.
a. Sure, let's call the time since it started bouncing "t" and the height of the weight as "h". If the weight starts at a height of 10 cm and reaches a maximum height of 50 cm at 1.5 s, we can model its height using a sine function. The equation for the height as a function of time can be given as:
h(t) = A*sin(B*t + C) + D
In this case, the amplitude (A) is the difference between the maximum and minimum heights, which is 50 cm - 10 cm = 40 cm. The period (B) is 2π divided by the time it takes to complete one full cycle, which is 3 s. The phase shift (C) is 0 since we are not shifting the starting point. The vertical shift (D) is the initial height, which is 10 cm. So the equation becomes:
h(t) = 40*sin((2π/3)*t) + 10
b. Unfortunately, as a bot, I'm unable to graph the equation directly. However, if you input the equation into a graphing tool, it will show you the corresponding graph.
c. The weight is moving up fastest when its height is increasing the fastest. This occurs at the point where the slope of the graph is maximum, which corresponds to the maximum or minimum points of the sine function. In this case, the first maximum height of 50 cm occurs at 1.5 s, so the weight is moving up fastest at that time.
d. Similarly, the weight is moving down fastest when its height is decreasing the fastest. This also occurs at the points where the slope of the graph is minimum, which are the midpoints between the maximum and minimum points of the sine function. In this case, the weight reaches its highest point at 50 cm at 1.5 s, which is the maximum point. It would start moving down fastest at the midpoint between the maximum and minimum heights, which would be when it's at a height of (50 cm + 10 cm) / 2 = 30 cm.
e. The weight changes direction when it reaches either its maximum or minimum height. So, the weight would change direction at the maximum point, which occurs at 1.5 s in this case.
To answer these questions, we need to analyze the given information and use it to derive the equation and understand the behavior of the weight attached to the spring.
a. Write an equation for the height, in cm, of the weight as a function of time, in seconds:
Based on the given information, the height of the weight can be modeled by a sine function. The general equation for a sine function is:
y = A * sin(B * (x - C)) + D
Where:
- A represents the amplitude, which is half the difference between the maximum and minimum height. In this case, since the weight reaches a maximum height of 50 cm and starts from a minimum height of 10 cm, A = (50 - 10) / 2 = 20 cm.
- B represents the frequency of the oscillation, which is related to the time it takes to complete one full cycle. Since the weight reaches its first maximum height at 1.5s, we can use this information to find B. One full cycle corresponds to the time it takes for the weight to go from its highest point, to its lowest, and back to its highest point. Therefore, the time for one cycle is 3s (2 * 1.5s). And since one complete cycle corresponds to a period of 2π, B = (2π) / 3.
- C represents the phase shift, which determines where the wave starts. In this case, no phase shift is mentioned, so C = 0.
- D represents the vertical shift, which determines the baseline height of the oscillation. In this case, D = 10 cm (the minimum height).
Therefore, the equation for the height of the weight as a function of time is:
y = 20 * sin((2π/3) * x) + 10
b. Graph the equation from part a:
To graph the equation, plot points for different values of x (time) and calculate the corresponding values of y (height). Then, connect these points to create the graph.
c. When is the weight moving up fastest:
The weight is moving up fastest when the slope of the height function is steepest. In this case, the weight is moving up fastest at the maximum points of the sine wave. Using the equation y = 20 * sin((2π/3) * x) + 10, the maximum points occur at x = 1.5s and x = 4.5s (adding an additional period). At these points, the weight is moving up fastest.
d. When is it moving down faster:
Similarly, the weight is moving down faster when the slope of the height function is steepest, which occurs at the minimum points of the sine wave. Using the equation y = 20 * sin((2π/3) * x) + 10, the minimum points occur at x = 0s, x = 3s, x = 6s, and so on. At these points, the weight is moving down faster.
e. At what time is the weight changing direction:
The weight changes direction (from moving up to moving down or vice versa) when the height function crosses the x-axis, meaning y = 0. To find these points, we can set the equation y = 20 * sin((2π/3) * x) + 10 equal to 0 and solve for x.