An aeroplane flying 15km due north and later changes direction and fly 5km due west.
1: Calculate the magnitude of the aeroplane.
2: Calculate the direction.
To calculate the magnitude of the airplane's displacement, we can use the Pythagorean theorem. The theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.
In this case, the airplane has flown 15 km due north and 5 km due west. We can represent these distances as the two sides of a right-angled triangle.
Let's label the 15 km side as the north side and the 5 km side as the west side. The hypotenuse of the triangle represents the magnitude of the airplane's displacement.
Using the Pythagorean theorem, we can calculate the magnitude (M) as follows:
M^2 = (15 km)^2 + (5 km)^2
M^2 = 225 km^2 + 25 km^2
M^2 = 250 km^2
Taking the square root of both sides, we find:
M = √250 km
M ≈ 15.81 km
Therefore, the magnitude of the airplane's displacement is approximately 15.81 km.
To calculate the direction of the airplane’s displacement, we can use trigonometry. Since the airplane initially flew due north and then due west, the direction will be the angle between the north direction and the direction of the airplane's displacement.
In this case, the angle can be found using the tangent function:
tan(θ) = Opposite / Adjacent
tan(θ) = 5 km / 15 km
θ = tan⁻¹(5/15)
Using a calculator, we find:
θ ≈ 18.43 degrees
Therefore, the direction of the airplane's displacement is approximately 18.43 degrees west of north.