A plane if flying at 200 mph with a heading of 45 degrees and encounters a wind of 100 mph from the west. What is the resulting velocity and heading?
East speed = 100 + 200 sin 45 = 241
North speed = 200 cos 45 = 141
speed = sqrt( 141^2 + 241^2) = 279 mph
tan east of north = 241/141
so angle east of north = 60 degrees
To find the resulting velocity and heading of the plane, we can use vector addition.
First, let's break down the velocities into their components. The velocity of the plane can be broken down into its horizontal (east-west) and vertical (north-south) components. The horizontal component can be calculated using the cosine of the angle of 45 degrees, and the vertical component can be calculated using the sine of the angle.
Horizontal component of the plane's velocity = 200 mph * cos(45 degrees)
Vertical component of the plane's velocity = 200 mph * sin(45 degrees)
The wind is blowing from the west, which means it will only affect the horizontal component of the plane's velocity and not the vertical component. Hence, the horizontal component of the wind's velocity will be 100 mph in the negative direction (west) since it is against the plane's direction of flight.
Now, let's add the horizontal components of the plane's velocity and the wind's velocity to find the resulting horizontal component:
Resulting horizontal velocity = Horizontal component of plane's velocity + Horizontal component of wind's velocity
= (200 mph * cos(45 degrees)) + (-100 mph)
= 141.42 mph - 100 mph
= 41.42 mph eastward
The vertical component of the plane's velocity remains unaffected by the wind. Hence, the vertical component of the resulting velocity will be the same as the vertical component of the plane's velocity:
Resulting vertical velocity = Vertical component of plane's velocity
= 200 mph * sin(45 degrees)
= 141.42 mph northward
Finally, we can calculate the magnitude and direction of the resulting velocity using the Pythagorean theorem and inverse tangent:
Magnitude of resulting velocity = sqrt(Resulting horizontal velocity^2 + Resulting vertical velocity^2)
= sqrt((41.42 mph)^2 + (141.42 mph)^2)
≈ 148.44 mph
Direction of resulting velocity = atan(Resulting vertical velocity / Resulting horizontal velocity)
= atan(141.42 mph / 41.42 mph)
≈ 73.54 degrees
Therefore, the resulting velocity is approximately 148.44 mph with a heading of approximately 73.54 degrees.