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
a) We can find the time at which the child reaches her next maximum and minimum heights by using the formula for the period of a pendulum. The period (T) is the time it takes for the child to complete one full swing. In this case, the child reaches maximum height, minimum height, and then maximum height again, so the period is the time from reaching a maximum height to reaching the next maximum height.
The given information tells us that the swing takes 0.6 seconds to reach the first maximum height, and then 1.3 seconds later (i.e. 0.6 + 1.3 = 1.9 seconds after starting), the child reaches a minimum height.
Since the period (T) is the time from a maximum height to the next maximum height, we can calculate it by subtracting the time it takes to reach a minimum height from the time it takes to reach the first maximum height:
T = 1.9 seconds - 0.6 seconds = 1.3 seconds
Therefore, the child will reach the next maximum height after 1.3 seconds from reaching the first maximum height (i.e. 0.6 + 1.3 = 1.9 seconds after starting).
To find the times for the next two maximum heights, we can add one period (1.3 seconds) to the previously calculated time:
1st Maximum Height: 0.6 seconds
2nd Maximum Height: 0.6 + 1.3 seconds = 1.9 seconds
3rd Maximum Height: 1.9 + 1.3 seconds = 3.2 seconds
4th Maximum Height: 3.2 + 1.3 seconds = 4.5 seconds
To find the times for the next two minimum heights, we can add half of a period (0.65 seconds) to the previously calculated time:
1st Minimum Height: 0.6 + 1.3 + 0.65 seconds = 2.55 seconds
2nd Minimum Height: 2.55 + 0.65 seconds = 3.2 seconds
3rd Minimum Height: 3.2 + 0.65 seconds = 3.85 seconds
4th Minimum Height: 3.85 + 0.65 seconds = 4.5 seconds
b) The equation for a sinusoidal function represents the height of the child on the swing at any given time. The general equation for a sinusoidal function is:
y = A * sin(B(x - C)) + D
where A is the amplitude, B is the period, C is the phase shift, and D is the vertical shift.
The amplitude (A) is half the difference between the maximum and minimum heights, which is (2.4 - 0.8) / 2 = 0.8 meters.
The period (T) is the time it takes for the child to complete one full swing, which we found to be 1.3 seconds.
The phase shift (C) determines where the graph starts. It is equal to the time at the first maximum height, which is 0.6 seconds.
The vertical shift (D) determines how the graph is shifted up or down. In this case, the minimum height is 0.8 meters, so D = 0.8.
Therefore, the equation representing the child on the swing is:
y = 0.8 * sin((2π/1.3)(x - 0.6)) + 0.8
c) To check our function, we can substitute the times we calculated in part a into the equation and see if it matches the given heights.
At t = 0.6 seconds:
y = 0.8 * sin((2π/1.3)(0.6 - 0.6)) + 0.8
y = 0.8 * sin(0) + 0.8
y = 0.8 + 0.8
y = 1.6 meters (matches the given maximum height of 2.4 meters)
At t = 1.9 seconds:
y = 0.8 * sin((2π/1.3)(1.9 - 0.6)) + 0.8
y = 0.8 * sin((2π/1.3)(1.3)) + 0.8
y ≈ 0.8 * sin(3.8466) + 0.8
y ≈ 0.8 * 0.1668 + 0.8
y ≈ 0.13344 + 0.8
y ≈ 0.93344 meters (approximately matches the given minimum height of 0.8 meters)
We can continue to check the other times calculated in part a to verify that the function matches the given heights.