Two radar stations A and B, are separated by horizontal distance of (a=500 metres), they both track the position of a plane C by recording angles (alpha), and (beta) at 1 second intervals.
time (sec) 9 10 11
the angle A(alpha) 54.80 54.06 53.34
the angle B (beta) 65.59 64.59 63.62
Note: angles given in degrees
If three successive readings are shown in the table
calculate the speed v of the plane and the climb angle at time t=10 seconds.
the coordinates of the plane can be shown to be as follows,
x =a*(tan(beta))/(tab(beta)- tan(alpha))
y =a*((tan(alpha)*tan(beta))/(tan(beta)-tan(alpha))
here x and y give the normal coordinates
from the first station. we need to find a function in terms of (alpha,Beta)by using calculus then substitute into this
values of alpha,beta when t=10 sec.
To calculate the speed (v) of the plane and the climb angle at time t=10 seconds, we need to find a function in terms of angles (alpha) and (beta) using calculus, and then substitute the values of (alpha) and (beta) when t=10 seconds.
Let's start by finding the function:
1. The speed (v) of the plane can be calculated using the formula:
v = a / (tab(t) - tan(alpha))
Where a is the horizontal distance between the radar stations (a = 500 meters) and tab(t) is the difference in the tangents of angles (alpha) and (beta) measured at time t.
2. The climb angle can be calculated using the formula:
tan(climb angle) = (tan(alpha) * tan(beta)) / (tan(beta) - tan(alpha))
Now, let's substitute the values of angles (alpha) and (beta) at time t=10 seconds into the formulas to find the speed (v) and the climb angle:
Given values:
- Angle A (alpha) at t=10 seconds: 54.06 degrees
- Angle B (beta) at t=10 seconds: 64.59 degrees
1. Speed (v) calculation:
v = a / (tab(t) - tan(alpha))
Substitute the respective values:
v = 500 / (tan(beta) - tan(alpha))
v = 500 / (tan(64.59) - tan(54.06))
v ≈ 500 / (2.21669 - 1.39897)
v ≈ 500 / 0.81772
v ≈ 611.13 meters/second
2. Climb angle calculation:
tan(climb angle) = (tan(alpha) * tan(beta)) / (tan(beta) - tan(alpha))
Substitute the respective values:
tan(climb angle) = (tan(54.06) * tan(64.59)) / (tan(64.59) - tan(54.06))
tan(climb angle) ≈ (1.39007 * 2.16467) / (2.16467 - 1.39007)
tan(climb angle) ≈ 3.00909 / 0.7746
tan(climb angle) ≈ 3.8844
To find the climb angle, take the arctan of 3.8844:
climb angle ≈ arctan(3.8844)
climb angle ≈ 76.24 degrees
Therefore, at time t=10 seconds:
- The speed (v) of the plane is approximately 611.13 meters/second.
- The climb angle is approximately 76.24 degrees.
To find the speed v of the plane and the climb angle at time t=10 seconds, we will use calculus to derive a function in terms of alpha and beta.
Let's start by finding the derivative of x with respect to beta, and the derivative of y with respect to beta:
dx/d(beta) = a * (sec^2(beta)) / (tan(beta) - tan(alpha))
dy/d(beta) = a * (tan(alpha) * sec^2(beta)) / (tan(beta) - tan(alpha))
Notice that we have used the derivative of tan(x) = sec^2(x) in the above steps.
Next, to find the derivative of x with respect to alpha, and the derivative of y with respect to alpha, we can apply the chain rule:
dx/d(alpha) = dx/d(beta) * d(beta)/d(alpha)
dy/d(alpha) = dy/d(beta) * d(beta)/d(alpha)
To find d(beta)/d(alpha), we can differentiate the equation tan(beta) - tan(alpha) = (y/x) with respect to alpha:
d(beta)/d(alpha) = -1 / (1 + tan^2(alpha))
Now, substituting the derivatives into the chain rule equations:
dx/d(alpha) = [a * (sec^2(beta)) / (tan(beta) - tan(alpha))] * [-1 / (1 + tan^2(alpha))]
dy/d(alpha) = [a * (tan(alpha) * sec^2(beta)) / (tan(beta) - tan(alpha))] * [-1 / (1 + tan^2(alpha))]
Finally, we can substitute the values of alpha and beta at t=10 seconds into these equations to find the derivatives at that time. Calculate dx/d(alpha) and dy/d(alpha) as follows:
dx/d(alpha) = [a * (sec^2(64.59)) / (tan(64.59) - tan(54.06))] * [-1 / (1 + tan^2(54.06))]
dy/d(alpha) = [a * (tan(54.06) * sec^2(64.59)) / (tan(64.59) - tan(54.06))] * [-1 / (1 + tan^2(54.06))]
Now that we have the derivatives, we can calculate the speed and climb angle at t=10 seconds:
v = sqrt((dx/d(alpha))^2 + (dy/d(alpha))^2)
climb angle = atan(dy/d(alpha) / dx/d(alpha))
Substitute the values we calculated above into these equations to find the speed and climb angle.