A jet airliner moving at 500 mph due east moves into a region where the wind is blowing at 120 mph in a direction north . What is the new velocity and direction of the aircraft?
Perform a vector addition
500 i + 120 j has a magnitude of
sqrt(500^2 + 120^2) = 514.2 mph
The ration of east and north components tells you the tangent of the direction.
To find the new velocity and direction of the aircraft, we need to combine the velocity of the jet with the velocity of the wind.
1. Convert the velocity of the jet and wind into vector form:
- The velocity of the jet is 500 mph due east, which can be written as (500, 0) mph.
- The velocity of the wind is 120 mph in a direction north, which can be written as (0, 120) mph.
2. Add the two vectors to find the resultant velocity:
- (500, 0) + (0, 120) = (500, 120) mph.
3. Calculate the magnitude and direction of the resultant velocity:
- The magnitude of the resultant velocity is found using the Pythagorean theorem:
magnitude = √(500^2 + 120^2) ≈ 519.62 mph.
- The direction of the resultant velocity can be found using trigonometry:
direction = arctan(120/500) ≈ 13.9 degrees north of east.
Therefore, the new velocity of the aircraft is approximately 519.62 mph at a direction of 13.9 degrees north of east.
To determine the new velocity and direction of the aircraft, we need to combine its velocity (due east) with the wind's velocity (north).
First, let's break down the velocities into their components.
The aircraft's velocity of 500 mph due east can be broken down as follows:
- Velocity in the eastward direction (x-axis): 500 mph
- Velocity in the northward direction (y-axis): 0 mph, since it is moving due east and not changing its altitude.
The wind's velocity of 120 mph in the north direction can be broken down as follows:
- Velocity in the eastward direction (x-axis): 0 mph, since the wind is not blowing from the east.
- Velocity in the northward direction (y-axis): 120 mph.
Now, we can add the two velocities together component-wise to find the resulting velocity:
- The resulting velocity in the eastward direction (x-axis) will be the sum of the aircraft's eastward velocity and the wind's eastward velocity. In this case, it will be 500 mph + 0 mph = 500 mph.
- The resulting velocity in the northward direction (y-axis) will be the sum of the aircraft's northward velocity and the wind's northward velocity. In this case, it will be 0 mph + 120 mph = 120 mph.
Therefore, the new velocity of the aircraft is 500 mph due east and 120 mph north.
To state it more succinctly, the new velocity of the aircraft is approximately 500 mph at an angle of 14.5 degrees north of east, assuming the aircraft maintains a constant altitude.