Homework Problem:Velocities have both direction and magnitude and so are vectors. The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N45W at a speed of 50 km/h. (This means that the direction from which the wind blows is 45 degrees west of the northly direction.) A pilot is steering a plane in the direction N60E at an airspeed (speed in still air) of 250 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. Use maple to find the true course and the ground speed of the plane.AnswerWe first enter the two vectors to maple. The wind velocity vector is: |
> wind_vel := vector([ -50*sin(Pi/4), 50*cos(Pi/4) ]);
1/2 1/2
wind_vel := [ - 25 2 , 25 2 ]
| and the plane's airspeed is entered by first converting 60 degrees to radians, |
> deg60 := convert(60*degrees,radians);
deg60 := 1/3 Pi
| and then entering the components of the vector: |
> airspeed := vector([ 250*sin(deg60), 250*cos(deg60) ]);
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airspeed := [ 125 3 , 125 ]
| the true course is then: |
> track := evalm(wind_vel + airspeed);
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track := [ - 25 2 + 125 3 , 25 2 + 125 ]
| which is approximately equal to, |
> evalf(evalm(track),5);
[ 181.16, 160.36 ]
| i.e. about 181 km/h East and 160 km/h North. The magnitude of the track gives the ground speed. |
> ground_speed := norm(evalm(track),2);
1/2 1/2 2 1/2 2 1/2
ground_speed := ((- 25 2 + 125 3 ) + (25 2 + 125) )
241.94
| i.e. about 242 km/h. |