The magnitude of a velocity vector is called speed. Suppose that a wind is blowi

Question

The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N45^degree W at a speed of 60 km/h. (This means that the direction from which the wind blows is 45^degree west of the northerly direction.) A pilot is steering a plane in the direction N60^degree E at an airspeed (speed in still air) of 150 km/h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane. (Round your answers to one decimal place.)

Explanation / Answer

This is a classical case of Vector Addition, where the quantity is Velocity.

Suppose the speed of wind (magnitude of Velocity vector of wind) to be A, which is 60 km/hr as given in the problem.

Suppose the speed (speed in still air) of Plane be B, which is 150 km/hr as given in the problem.

Vector A (wind) blows in N45o W direction and Vector B (Plane) flies in N60OE direction.

Using the formula for Vector Addition,

Magnitude Resultant, R = (A2+B2+2AB Cos €)1/2

where € = angle between the vectors

In this case € = 45o+60o = 105o

So applying the above formula,

Magnitude of Resultant,R= (602+1502+ 2*60*150*Cos105o)1/2

R = 146.4 km/hr

It is the ground speed of the plane.

Direction of Resultant, @ = tan-1 ( B sin €/A + B cos €)

@ = tan-1 (150 sin 105o / 60 + 150 cos 105o)

@ = 81.6 o ( taking Vector A as reference)

So the true course of Plane is (81.6o - 45o) = 36.6o

So the true course is N 36.6E.

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