Calculating Total Distance and Direction of an Airplane

Question:

An airplane flies 20km in a direction 60 degrees north of east, then 30 km straight east, then 10km straight north. How far and in what direction is the plane from the starting point?

Answer:

48.4 km, 34.3° north of east

Let's break down the given data to calculate the total distance and direction of the airplane from the starting point. The airplane first travels 20km in a direction 60 degrees north of east, followed by 30 km directly east, and then 10km directly north.

To find the total distance traveled by the airplane, we need to calculate the x and y components of its motion. By summing up the x components (east direction) and y components (north direction) of the vectors, we can determine the total displacement.

Adding up the x components of the vectors:

x = 20 cos 60 + 30 + 0

x = 40 km

Adding up the y components of the vectors:

y = 20 sin 60 + 0 + 10

y = 27.3 km

The magnitude of the displacement is calculated using the Pythagorean theorem:

d = √(x² + y²)

d = 48.4 km

The direction of the airplane from the starting point is determined using the arctangent function:

θ = atan(y/x)

θ = 34.3° north of east

Therefore, the airplane is approximately 48.4 km away from the starting point at a direction of 34.3 degrees north of east.