1. a child on a swing reaches the maximum of height 2.4m, 0.6 seconds after starting their swing. 1.3s later they reach a minimum of 0.8m.
a) at what time, s, does she reach her next three MAX heights and MIN heights?
b) create a sinusoidal function that represents the child on this swing (equation)
c) check your function from part a
2. the child on the swing experiences double the amplitude and double the period.
a) what is the max and min height for the swinger now?
b) what is the equation of axis (axis of curve)
c) what is the new equation for the swinger?
d) what would you notice watching the swinger durning this change?
a) To find the times at which the child reaches her next three maximum and minimum heights, we need to find the period of the swing.
The time it takes for the child to go from the maximum height to the minimum height is 0.6s + 1.3s = 1.9s. This is one complete period of the swing.
Therefore, the next three maximum heights will occur at intervals of one period. Adding 1.9s to the initial time of 0.6s, we find:
First maximum height: 0.6s + 1.9s = 2.5s
Second maximum height: 2.5s + 1.9s = 4.4s
Third maximum height: 4.4s + 1.9s = 6.3s
Similarly, the next three minimum heights can be found at intervals of one period:
First minimum height: 0.6s + 1.9s = 2.5s
Second minimum height: 2.5s + 1.9s = 4.4s
Third minimum height: 4.4s + 1.9s = 6.3s
b) To create a sinusoidal function that represents the child on the swing, we can use the general form:
y = A*sin(B(x - C)) + D
Where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
From the given information, we can determine the values of A, B, C, and D:
Amplitude: The maximum height is 2.4m and the minimum height is 0.8m. So, the amplitude is (2.4 - 0.8)/2 = 0.8m.
Frequency: The period is 1.9s, so the frequency is 1/1.9 Hz.
Phase Shift: The initial time is 0.6s, so the phase shift is 0.6s.
Vertical Shift: The average height of the swing is (2.4 + 0.8)/2 = 1.6m.
Therefore, the equation for the child on the swing is:
y = 0.8*sin((2π/1.9)(x - 0.6)) + 1.6
c) To check the function from part a, we can substitute the values of x that correspond to the times at which the child reaches the maximum and minimum heights:
For the next three maximum heights: x = 2.5s, x = 4.4s, x = 6.3s
For the next three minimum heights: x = 2.5s, x = 4.4s, x = 6.3s
Plugging these values into the equation, we can calculate the corresponding y-values. If the values match the given maximum and minimum heights, then the equation is correct.
2. For this part, we are given that the child on the swing experiences double the amplitude and double the period.
a) The maximum height would be 2.4m * 2 = 4.8m, and the minimum height would be 0.8m * 2 = 1.6m.
b) The equation of the axis (axis of the curve) remains the same as it represents the vertical shift. So, it is still y = 1.6.
c) The new equation for the swinger can be found by doubling the amplitude and doubling the period in the previous equation:
y = 1.6 + 1.6*sin((2π/0.95)(x - 0.6))
d) Watching the swinger during this change, one would notice that the swing reaches higher max heights and lower min heights compared to before. The swing also takes longer to complete one full swing, resulting in a slower swinging motion.