Step-by-step explanation:
To calculate the velocity of the plane as a function of time, we need to compute the derivatives of the given polar coordinates, θ and r, with respect to time (t). We can use finite-difference methods to approximate these derivatives accurately.
There are several finite-difference methods available, such as forward difference, backward difference, and central difference. Each method has its advantages and limitations. In this case, we'll use the central difference method, which provides a more accurate approximation by considering neighboring data points on both sides.
Given data points:
t = [200, 202, 204, 206, 208, 210]
θ = [0.75, 0.72, 0.70, 0.68, 0.67, 0.66]
r = [5120, 5370, 5560, 5800, 6030, 6240]
To calculate the velocity V = √r^2 + r(θ)^2, we need to compute the derivatives dθ/dt and dr/dt.
1. Calculate dθ/dt (approximation):
To approximate dθ/dt, we'll use the central difference method. Since the time interval between consecutive points is 2 seconds, we can use the formula:
dθ/dt ≈ (θ[i+1] - θ[i-1]) / (t[i+1] - t[i-1])
For i = 1 to 4 (excluding the first and last data points), the central difference formula can be applied. For the first and last points, we'll use forward and backward difference respectively.
dθ/dt = [(θ[2] - θ[0]) / (t[2] - t[0]),
(θ[3] - θ[1]) / (t[3] - t[1]),
(θ[4] - θ[2]) / (t[4] - t[2])]
2. Calculate dr/dt (approximation):
Similar to dθ/dt, we'll use the central difference method to approximate dr/dt.
dr/dt ≈ (r[i+1] - r[i-1]) / (t[i+1] - t[i-1])
For i = 1 to 4 (excluding the first and last data points), the central difference formula can be applied. For the first and last points, we'll use forward and backward difference respectively.
dr/dt = [(r[2] - r[0]) / (t[2] - t[0]),
(r[3] - r[1]) / (t[3] - t[1]),
(r[4] - r[2]) / (t[4] - t[2])]
3. Calculate V (velocity):
Using the derived values of dθ/dt and dr/dt, we can calculate V = √(r^2 + r(θ)^2) using the given formula.
V = √(r^2 + r(θ)^2)
Please note that we are assuming uniform time intervals between the data points. Additionally, as with any finite-difference method, the accuracy of the approximation increases with smaller time intervals.