Total Displacement: Calculating the Sum of Translations

How do we determine the total displacement of an airplane's journey?

An airplane heads north of east by 18 degrees for a distance of 67 km, then heads due north for 39 km. What is the plane's total displacement?

a) 106 km

b) 74 km

c) 86 km

d) 109 km

The Total Displacement of the Airplane:

The total displacement of the airplane can be found by adding the vector components of each segment of the trip and calculating the magnitude of the resultant vector.

Explanation:

The student's question involves finding the airplane's total displacement after flying north of east by 18 degrees for a distance of 67 km, and then flying due north for 39 km. To calculate the displacement, we use vector addition. The displacement vector for the eastward and northward flight can be represented as two separate components that can be combined to find the resultant vector.

The eastward leg creates a horizontal component (67 km × cos(18°)) and a northward vertical component (67 km × sin(18°)). The second leg adds a purely northward component of 39 km to the vertical component.

Calculation:

Eastward component: 67 km × cos(18°) = 63.3 km (approx)

Northward component: 67 km × sin(18°) + 39 km = 21.2 km + 39 km = 60.2 km (approx)

The magnitude of the total displacement (hypotenuse): √(63.3² + 60.2²) ≈ 84.5 km

Therefore, the plane's total displacement is closest to 86 km, option c).