1. a child on a swing reaches the maximum of height 2.4m, 0.6 seconds after starting their swing. 1.3 seconds later the reach a minimum of 0.8m.
a) at what time, s, does she reach her next three maximum heights and minimum 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 maximum and minimum height for the swinger now?
b) what is the new equation of axis (axis of curve)
c) what is the new equation for the swinger?
d) what would you notice watching the swinger during this change?
a) To find the time at which the child reaches the next three maximum and minimum heights, we need to analyze the time intervals between the previous maximum and minimum heights.
From the given information, we know that the child reaches the maximum height of 2.4m, 0.6 seconds after starting their swing. Then, 1.3 seconds later, they reach a minimum height of 0.8m.
The time between the start of the swing and the first maximum height is 0.6 seconds.
The time between the first maximum height and the first minimum height is 1.3 seconds.
Therefore, the time taken for one complete cycle (one maximum to the next maximum or one minimum to the next minimum) is the sum of these two times: 0.6 + 1.3 = 1.9 seconds.
To determine the time at which the child reaches the next three maximum heights and minimum heights, we can apply this cycle time repeatedly.
b) The equation representing the child on the swing can be written as:
h(t) = A * sin(2π/T * (t - t0)) + C
where:
h(t) represents the height of the child at time t
A represents the amplitude of the swing
T represents the period of the swing (time for one complete cycle)
t0 represents the initial time when the swing is started
C represents the vertical shift (average height)
c) To check the function from part a, we need to plug in specific values and see if they match the given data.
For example, let's say the first maximum height is reached at t = 0.6s, and the height at that time is 2.4m.
Plugging these values into the equation, we can solve for A, T, t0, and C:
2.4 = A * sin(2π/T * (0.6 - t0)) + C
Similarly, we can use the second known data point:
0.8 = A * sin(2π/T * (1.9 - t0)) + C
Using these two equations, we can solve for A, T, t0, and C to check if the function matches the given data.
2. For this question, the amplitude and period of the swing have doubled.
a) The maximum height for the swinger now will be double the previous amplitude, so it will be 2 * 2.4 = 4.8m.
The minimum height will also be double the previous amplitude, so it will be 2 * 0.8 = 1.6m.
b) The equation of the axis (axis of the curve) remains the same, as it represents the average height of the swinger. So it is C = (2.4 + 0.8) / 2 = 1.6m.
c) The new equation for the swinger can be written as:
h(t) = 4.8 * sin(2π/(2 * 1.9) * (t - t0)) + 1.6
where the amplitude is doubled (4.8), and the period is doubled (2 * 1.9). The axis of the curve is still at 1.6m.
d) If you were watching the swinger during this change, you would notice that the swinger swings higher and takes more time to complete one full cycle (from maximum to maximum or minimum to minimum). The motion will appear slower and more elongated compared to the previous swing.