A plane flying 16 degrees west of north at 289mph is flying in air which is moving 32mph at due west. Find the resulting speed and direction of the plane. (Step by step please!)
I would just add up the two vectors
(289cos106°, 289sin106°) + (32cos180, 32sin180)
= (-79.659 , 277.805) + (-32 , 0)
= (-109.659 , 277.805)
magnitude = √(109.659^2 + 277.805^2)
= 298.665 mph
direction ?
tanØ = 277.805/-109659
Ø = 111.54°
which can be expressed as N 21.5° W
A plane is heading 25° west of south. How many degrees south of west is this plane flying?
To find the resulting speed and direction of the plane, we need to consider the vector addition of the plane's velocity and the velocity of the air.
Step 1: Break down the plane's velocity into its north and west components.
The plane is flying 16 degrees west of north, so we can represent this as an angle between the plane's velocity vector and the north direction.
Let's assume that the plane's velocity magnitude (speed) is Vp, which is 289 mph in this case.
The west component of the plane's velocity (Vp_west) can be calculated using trigonometry:
Vp_west = Vp * sin(16 degrees)
The north component of the plane's velocity (Vp_north) can also be calculated with trigonometry:
Vp_north = Vp * cos(16 degrees)
Step 2: Determine the velocity of the air.
The velocity of the air is given as 32 mph to the west. Since it is moving in the west direction only, there is no northward component to consider.
So the air's velocity would be V_a = 32 mph in the west direction.
Step 3: Add the north and west components of the plane's velocity vector and the velocity of the air vector to get the resulting velocity.
The resulting westward component (V_resultant_west) can be found by adding the westward components of the plane and air velocities:
V_resultant_west = Vp_west + Va = Vp * sin(16 degrees) + 32 mph
The resulting northward component (V_resultant_north) can be found by subtracting the northward component of the plane's velocity from the northward component of the air velocity:
V_resultant_north = Vp_north - 0 mph (since the air is not moving northward)
Step 4: Calculate the magnitude of the resulting velocity.
The magnitude of the resulting velocity (V_resultant) can be found using the Pythagorean theorem:
V_resultant = sqrt((V_resultant_west)^2 + (V_resultant_north)^2)
Step 5: Calculate the direction of the resulting velocity.
The direction of the resulting velocity can be determined using inverse trigonometric functions:
θ = arctan(V_resultant_north / V_resultant_west)
Finally, we have the resulting speed and direction of the plane.