An airplane is heading due south at a speed of 515 km/h. A wind begins blowing from the southwest at a speed of 100 km/h (average). (Hint: First draw a diagram.)
(a) What is the velocity (magnitude and direction) of the plane relative to the ground? magnitude/direction (east of south)
(b) How far off course will it be after 8 min if the pilot takes no corrective action?
225
a,515km/h-100km/h=415km/h
i.e (415km/h*1000m)/1h(60*60)s=115m/s
8km
To solve this problem, we need to break down the velocity of the airplane into its components and then account for the effect of the wind.
(a) To find the velocity of the plane relative to the ground, we need to calculate the resultant velocity vector. First, draw a diagram to visualize the situation. Assume the direction towards the south is positive.
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Wind |__________/ Ground
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Airplane
The velocity of the airplane can be broken down into two components: a north-south component and an east-west component.
Since the airplane is heading due south, the magnitude of its velocity in the north-south direction is 515 km/h.
The wind is coming from the southwest, which is 45 degrees west of south. To find the east-west component of the wind, we need to calculate the cosine of 45 degrees and multiply it by the wind speed: cos(45°) = √2/2 ≈ 0.707.
So, the magnitude of the wind's east-west component is 0.707 × 100 km/h ≈ 70.7 km/h.
Now, we can find the resultant velocity of the airplane relative to the ground. We add the north-south component of the airplane's velocity to the east-west component of the wind's velocity:
North-south component: 515 km/h (towards the south)
East-west component: -70.7 km/h (opposite to the east)
To calculate the resultant velocity, we can use vector addition. Given the magnitudes and directions, we can use the Pythagorean theorem to find the magnitude and trigonometry to find the direction.
Magnitude of resultant velocity = √(515 km/h)^2 + (-70.7 km/h)^2 ≈ 520 km/h
Direction of resultant velocity = arctan((-70.7 km/h)/(515 km/h)) east of south
So, the velocity of the plane relative to the ground is approximately 520 km/h east of south.
(b) To find how far off course the plane will be after 8 minutes, we need to calculate the horizontal distance traveled by the plane, considering the wind's effect.
The horizontal distance traveled by the plane is equal to the east-west component of its velocity multiplied by the time:
Horizontal distance = (-70.7 km/h) × (8/60) h (since 8 minutes = 8/60 hours)
Horizontal distance ≈ -9.4 km
Since the east-west component is negative, it means the plane is being pushed to the west by the wind. Therefore, after 8 minutes, the plane will be about 9.4 km off course to the west.
Note: By convention, positive distances are measured to the east, and negative distances are measured to the west.