Question

Plane Ride
You are traveling on an airplane. The velocity of the plane with respect to the air is 110 m/s due east. The velocity of the air with respect to the ground is 43 m/s at an angle of 30° west of due north.
1)What is the speed of the plane with respect to the ground?
2)What is the heading of the plane with respect to the ground? (Let 0° represent due north, 90° represents due east).° East of due North
3)How far east will the plane travel in 1 hour?

Answer 1

this is a classic vector addition problem.
The first thing I do here is break these into components. We are given a specific grid to use in this case, so we will impose that on our graph and work from that...

Answer 2

we are told that the up direction is zero degrees, and right (east) is positive, so we have\[\overrightarrow{V_p}=|V_p|\cos\theta\overrightarrow{i}+|V_p|\sin\theta\overrightarrow{j}=110\cos0^{\circ}\overrightarrow{i}+110\sin0^{\circ}\overrightarrow{j}\]for the planes velocity vector, and\[\overrightarrow{V_a}=|V_a|\cos\theta\overrightarrow{i}+|V_a|\sin\theta\overrightarrow{j}=43\cos(-30^{\circ})\overrightarrow{i}+43\sin(-30^{\circ})\overrightarrow{j}\]The total velocity will be given by the vector sum of these two velocities\[\overrightarrow{V}=\overrightarrow{V_p}+\overrightarrow{V_a}\]The magnitude of this vector is its length, which can be found using the pythagorean theorem with the components\[|\overrightarrow{V}|^2=i^2+j^2\]so that will be your final speed.
The angle can be found by the inverse tangent of its components\[\theta=\tan^{-1}(\frac i j)\]so that will be the heading.
The distance the plane travels can be found by multiplying the final vectors horizontal speed component (in this case the j-component) by time.
Just make sure to represent an hour as 3600 seconds to keep units consistent.

Answer 3

here's a little sketch to remind us how the base components i and j go in this case