Question

a bird flies south at 31 mph, while the wind blows at an angle of 39° east of north, at 12 mph. what is the magnitude of the birds resultant vector?
draw a vector diagram.
|→r| = ? mph
round your answer to the nearest hundredth.

Explanation:

Step1: Define Vectors

Let the bird's velocity vector be \(\vec{v}_b\) (south, so \(y\)-component negative) and wind's velocity vector be \(\vec{v}_w\).
\(\vec{v}_b\): magnitude \(31\) mph, direction south (\(0\) east-west, \(-31\) north-south).
\(\vec{v}_w\): magnitude \(12\) mph, angle \(39^\circ\) east of north. So:
East component: \(v_{w,x} = 12\sin(39^\circ)\)
North component: \(v_{w,y} = 12\cos(39^\circ)\)

Step2: Combine Components

Resultant \(x\)-component (\(R_x\)): \(v_{w,x} + 0 = 12\sin(39^\circ)\)
Resultant \(y\)-component (\(R_y\)): \(v_{w,y} - 31 = 12\cos(39^\circ) - 31\)

Step3: Calculate Magnitude

Use \(|\vec{R}| = \sqrt{R_x^2 + R_y^2}\)
First, compute \(R_x\): \(12\sin(39^\circ) \approx 12 \times 0.6293 \approx 7.5516\)
\(R_y\): \(12\cos(39^\circ) - 31 \approx 12 \times 0.7771 - 31 \approx 9.3252 - 31 = -21.6748\)
Then, \(|\vec{R}| = \sqrt{(7.5516)^2 + (-21.6748)^2}\)
\((7.5516)^2 \approx 57.026\), \((-21.6748)^2 \approx 469.798\)
Sum: \(57.026 + 469.798 = 526.824\)
Square root: \(\sqrt{526.824} \approx 22.95\)

Answer:

\(22.95\)