Answer:
R = 207.45 mm , θ_return = 18.47 south west
Explanation:
This vector addition exercise is schematized in the attachment where the displacements are
1 d1 = 120 mm west
2 d2 = 250mm at 45 south east
3 d3 = 280 mm at 30 east of nort.
R is the final displacement that takes the goat to its initial point (origin)
The analytical way to perform this exercise is to find the components of each displacement and add them
Decompose the displacement using trigonometry
Displacement d1
d1ₓ = 120 cos 180 = -120 mm
Displacement d2, with the angle measured from the axis this θ = 270 + 45
sin 45 = [tex]d2_{y}[/tex] / d2
cos 45 = d2ₓ / d2
[tex]d2_{y}[/tex] = d2 sin45
[tex]d2_{y}[/tex] = 250 sin (270 + 45)
[tex]d2_{y}[/tex] = -176.77 mm
d2ₓ = d2 cos (270 + 45)
d2ₓ = 176.77 mm
displacement d3, for half the angle from the east axis θ = 90-30 = 60
sin 60 = [tex]d3_{y}[/tex] / d3
cos 60 = d3ₓ / d3
[tex]d3_{y}[/tex] = d3 sin 60
d3ₓ = d3 cos 60
[tex]d3_{y}[/tex] = 280 sin 60 = 242.49 mm
d3ₓ = 280 cos 60 = 140 mm
Having all the displacement components we can find the total displacement
Rₓ = d1ₓ + d2ₓ + d3ₓ
Ry = [tex]d1_{y}[/tex] + [tex]d2_{y}[/tex] + [tex]d3_{y}[/tex]
Rₓ = -120 + 176.77 +140
Rₓ = 196.77 mm
Ry = 0 -176.77 +242.49
Ry = 65.72 mm
Therefore the displacement you must make to return to the starting point is
R = RA Rx2 + Ry2)
R = RA (196.77 2 + 65.72 2)
R = 207.45 mm
We used trigonometry
tan tea = RY / Rx
tea = tan-1 Ry / Rx
ea = tan-1 (65.72 / 196.77)
tea = 18.47
This is the point where the girl is, to return to its origin this path must be serial, but in the opposite direction,
θ_return = 18.47 south west
