Question

A crow flies 23 km at a bearing of 043°. It then flies 47 km due south and lands. What is the displacement of the crow from its starting point?

Answer 1

The displacement of the crow from its starting point is 35.51 km.

Step-by-step explanation:

To find the displacement of the crow from its starting point, we first need to break down the crow's movement into its north-south and east-west components. We can use trigonometry to find these components.

We know that the crow flies 23 km at a bearing of 043°, which means it is moving in a direction of 43° east of north. Using trigonometry, we can find that the north-south component is 23 km * sin(43°) = 15.65 km and the east-west component is 23 km * cos(43°) = 16.82 km.

Next, the crow flies 47 km due south. This means its north-south component will decrease by 47 km, while its east-west component remains the same. So, the new north-south component is 15.65 km - 47 km = -31.35 km.

Finally, we can use the Pythagorean theorem to calculate the displacement of the crow. The displacement is the square root of the sum of the squares of the components. So, the displacement is √((-31.35 km)^2 + 16.82 km^2) = √(980.6 km^2 + 282.3 km^2) = √(1262.9 km^2) = 35.51 km.

Answer 2

To calculate the crow's displacement, breaking down its flight into northeast and due south components and determining the vector sum of these movements through trigonometric functions and the Pythagorean theorem is necessary.

Step-by-step explanation:

To find the displacement of the crow from its starting point, we need to use trigonometry to calculate the resultant vector from the two stages of its journey. In the first stage, the crow flies with a bearing of 043° for 23 km, which means it flies northeast. In the second stage, the crow flies 47 km due south.

To determine the displacement, we will resolve the northeast flight into north and east components and then combine the south component from the second leg. The northward component of the first leg can be calculated using cosine: 23 km × cos(043°), and the eastward component using sine: 23 km × sin(043°). We then subtract the second leg (47 km due south) from the northward component of the first leg.

The total displacement to the north is then given by 23 km × cos(043°) - 47 km, and the displacement to the east is 23 km × sin(043°). The overall displacement is the vector sum of these components, which can be found by applying the Pythagorean theorem. Finally, we determine the bearing angle of the displacement vector with respect to the north using the arctangent function.