A plane travelling at a speed of 150 meter per second due east of north has to go from point A to point B 500 km apart. If a wind blows straight north at a speed of 20 meter per second find the angle the pilot should follow to reach point B and also the time taken in flying from A to B.
The resultant velocity is
v = (106,106) + (0,20) = (106,126)
|v| = 164.7
The angle x obeys tan(x) = 126/106
The time taken is 500,000/164.7 seconds
To find the angle the pilot should follow and the time taken to fly from point A to B, we can break down the velocities and distances involved.
Let's first calculate the component of the plane's velocity due to the wind. Since the wind is blowing straight north and the plane is traveling due east of north, we can use vector addition to find the resultant velocity.
Given:
- Plane's velocity (Vp): 150 m/s due east of north
- Wind velocity (Vw): 20 m/s due north
To find the resultant velocity (Vr), we can use the Pythagorean theorem:
Vr^2 = Vp^2 + Vw^2
Substituting the given values:
Vr^2 = (150^2) + (20^2)
Vr^2 = 22500 + 400
Vr^2 = 22900
Vr ≈ 151.23 m/s
Now, we can find the angle the pilot should follow. Since the plane's velocity is directed due east of north, we need to find the angle between the resultant velocity (Vr) and the northerly direction.
Using trigonometry:
cos θ = Vw / Vr
Substituting the given values:
cos θ = 20 / 151.23
θ ≈ cos^(-1)(20 / 151.23)
θ ≈ 7.52 degrees
Therefore, the pilot should follow an angle of approximately 7.52 degrees to reach point B.
Next, let's calculate the time taken to fly from point A to point B.
Given:
- Distance to be covered (d): 500 km
- Plane's velocity (Vp): 150 m/s due east of north
First, convert the distance from kilometers to meters:
d = 500 km * 1000 m/km
d = 500,000 m
Now, divide the distance by the plane's velocity to find the time taken:
Time (t) = Distance / Velocity
t = 500,000 m / 150 m/s
t ≈ 3333.33 s
Therefore, the time taken to fly from point A to point B is approximately 3333.33 seconds.