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 during this change?
a) To find the time at which the child reaches the next three maximum and minimum heights, we can use the formula for the time period (T) of a pendulum swing:
T = 2 * π * (√(L/g))
where L is the length of the swing's string and g is the acceleration due to gravity.
Given that the maximum height is 2.4m, it occurs 0.6 seconds after starting the swing. This means that at t = 0.6 seconds, the child is at the maximum height.
Next, we need to find the period of the swing. To do this, we can use the fact that the total time for one complete swing (from maximum height to maximum height) is the sum of the time taken from maximum height to minimum height and from minimum height back to maximum height.
Given that the maximum height occurs 0.6 seconds after starting the swing and the minimum height occurs 1.3 seconds after the maximum height, the time for one complete swing is 0.6 seconds + 1.3 seconds = 1.9 seconds.
Now we can calculate the time at which the child reaches the next three maximum heights and minimum heights. The time for one complete swing is 1.9 seconds, so the time for one-half of a swing (from maximum height to maximum height or from minimum height to minimum height) is half of 1.9 seconds, which is 0.95 seconds.
Next Three Maximum Heights:
- The child reaches the first maximum height at t = 0.6 seconds.
- The child reaches the second maximum height at t = 0.6 seconds + 0.95 seconds = 1.55 seconds.
- The child reaches the third maximum height at t = 0.6 seconds + 2 * 0.95 seconds = 2.5 seconds.
Next Three Minimum Heights:
- The child reaches the first minimum height at t = 0.6 seconds + 1.3 seconds = 1.9 seconds.
- The child reaches the second minimum height at t = 0.6 seconds + 1.3 seconds + 0.95 seconds = 2.85 seconds.
- The child reaches the third minimum height at t = 0.6 seconds + 1.3 seconds + 2 * 0.95 seconds = 3.8 seconds.
b) To create a sinusoidal function that represents the child on this swing, we can use the formula:
h(t) = A * sin(2π/T * t) + D
where h(t) is the height at time t, A is the amplitude, T is the period, and D is the vertical displacement (or average height).
From the given information, the maximum height is 2.4m, so we can use A = 2.4m/2 = 1.2m (half of the maximum height). The period is 1.9 seconds, so T = 1.9 seconds. The average height is (2.4m + 0.8m)/2 = 1.6m.
Therefore, the sinusoidal function that represents the child on this swing is:
h(t) = 1.2 * sin(2π/1.9 * t) + 1.6
c) To check the function h(t) from part b, we can substitute the values of t from part a into the function and see if we get the corresponding heights.
For the first maximum height (t = 0.6 seconds):
h(0.6) = 1.2 * sin(2π/1.9 * 0.6) + 1.6
≈ 1.2 * sin(2π/1.14) + 1.6
≈ 1.2 * sin(5.4978) + 1.6
≈ 1.2 * 0.9870 + 1.6
≈ 2.3844 + 1.6
≈ 3.9844
The calculated height of 3.9844m matches the given maximum height of 2.4m.
Similarly, you can check the other heights from part a using the function h(t).
2. a) If the child on the swing experiences double the amplitude and double the period, the new maximum and minimum heights can be calculated as follows:
Original Maximum Height: 2.4m
Double the Amplitude: 2 * 2.4m = 4.8m
New Maximum Height = 4.8m + 0.8m (average height) = 5.6m
Original Minimum Height: 0.8m
Double the Amplitude: 2 * 0.8m = 1.6m
New Minimum Height = 1.6m + 0.8m (average height) = 2.4m
Therefore, the new maximum height is 5.6m and the new minimum height is 2.4m.
b) The equation of the axis (axis of the curve) represents the average height of the swing. In this case, the average height is 0.8m. So, the equation of the axis is h(t) = 0.8.
c) The new equation for the swinger can be obtained by doubling the amplitude and the period in the original function from part b:
New Amplitude = 2 * 1.2m = 2.4m
New Period = 2 * 1.9 seconds = 3.8 seconds
Therefore, the new equation for the swinger is:
h(t) = 2.4 * sin(2π/3.8 * t) + 0.8
d) When observing the swinger, you would notice that the new swing has larger peaks and deeper troughs compared to the original swing. The swing also takes longer to complete one full swing, resulting in a slower swinging motion.