How to Calculate Resultant Speed and Heading of an Airplane

Question: Given that a model airplane is flying north with a velocity of 15m/s and a strong wind blowing east at 12m/s, what is the airplane's resultant speed (magnitude of vector) and heading (direction of velocity vector)? Answer:
  1. The airplane's resultant speed is 19.21 m/s.
  2. The airplane's heading is 38.66 degrees east of north.

When dealing with vectors, it is important to consider both magnitude and direction. In the case of a model airplane flying north with a velocity of 15m/s and a strong eastward wind blowing at 12m/s, we can calculate the resultant speed and heading using trigonometry.

Resultant Speed Calculation:

To determine the resultant speed (magnitude of vector), we can use the Pythagorean theorem. The northward velocity of 15m/s and the eastward wind of 12m/s form a right triangle. By applying the theorem:

R = √(15² + 12²)

R = √(225 + 144)

R ≈ √369

R ≈ 19.21 m/s

Therefore, the airplane's resultant speed is approximately 19.21 m/s.

Heading Calculation:

To find the direction of the velocity vector (airplane's heading), we use the tangent function. The vertical and horizontal velocities act as the opposite and adjacent sides of a right triangle, respectively. Applying the tangent function:

θ = tan^-1(12/15)

θ ≈ 38.66 degrees

Thus, the airplane is heading approximately 38.66 degrees east of north.

By calculating both the resultant speed and the heading, we can have a clear understanding of the airplane's velocity in relation to its direction of movement. These calculations are essential for proper navigation and piloting of aircraft.

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