A crow flies in two successive displacements to a point that is 80 m to the west. Its first displacement is 80 m in a direction θ = 50° west of north.
What is the magnitude of its second displacement?
What is its direction as measured by the angle θ measured west from north?
8 years ago, Bob had no idea wtf he was talking about
Well, do the math:
Final position=displacement1+displcement2
80W=(80Sin50 W + 80 cos50N)+ displacement2
so I see displacement2 as
80(1-sin50)W -80 cos50 N or
18.7 W -51.2 N or 51.2S+18.7W
You can do the magnitude and direction
Would it be sqrt( (51.2)^2 + (18.7)^2 )?
To find the magnitude of the second displacement, we can use vector addition.
First, let's break down the first displacement into its north and west components. The magnitude of the first displacement is 80 m and the angle θ is 50° west of north.
To find the north component, we use the sine function:
North Component = 80 m * sin(θ)
North Component = 80 m * sin(50°)
North Component ≈ 61.03 m (rounding to two decimal places)
To find the west component, we use the cosine function:
West Component = 80 m * cos(θ)
West Component = 80 m * cos(50°)
West Component ≈ 51.34 m (rounding to two decimal places)
Now, since the crow flies to a point that is 80 m to the west, the second displacement will have a magnitude of 80 m as well.
To find the direction of the second displacement, we can use trigonometry.
Since the second displacement is to the west, the angle θ for the second displacement will be measured west from north.
To determine the angle west from north, we need to find the inverse tangent of the ratio of the second displacement's west component to its north component.
θ = tan^(-1)(West Component / North Component)
Substituting the values we calculated earlier, we have:
θ = tan^(-1)(51.34 m / 61.03 m)
θ ≈ tan^(-1)(0.8416)
θ ≈ 40.22° (rounding to two decimal places)
Therefore, the magnitude of the second displacement is 80 m and its direction, as measured by the angle θ measured west from north, is approximately 40.22° (rounding to two decimal places).