Answer:
R = 7,34 km
Ф = 63,43° north of east
Step-by-step explanation:
Let´s analyze each movement and find its perpendicular components. Then, we are going to add the components along the x and y axes to get the resultant and finally to calculate its magnitude and direction.
To determine the perpendicular components, we will use right triangles trigonometrics ratios.
Please see images for each displacement in attached file.
- <u>Displacement A: 2.50 km and 45° north of west</u>
Ax= AcosФ = 2.50cos45° = 1.77 km (neg. direction)
Ay= AsenФ = 2.50sen45° = 1.77 km (pos. direction)
- <u>Displacement B: 4.70 km and 60° south of east</u>
Bx= BcosФ = 4.70cos60° = 2,35 km (pos. direction)
By= BsenФ = 4.70sen60° = 4,07 km (neg. direction)
- <u>Displacement C: 1.30 km and 25° south of west </u>
Cx= CcosФ = 1.30cos25° = 1,18 km (neg. direction)
Cy= CsenФ = 1.30sen25° = 0.55 km (neg. direction)
- <u>Displacement D: 5.10 km due east (0°)</u>
Dx= DcosФ = 5.10 km (pos. direction)
Dy= DsenФ = 0 km
- <u>Displacement E: 1.70 km and 5° east of north</u>
Ex= EsenФ = 1.70sen5° = 0,15 km (pos. direction)
Ey= EcosФ = 1.70cos5° = 1.69 km (pos. direction)
- <u>Displacement F: 7.20 km and 55° south of west</u>
Fx= FcosФ = 7.20cos55° = 4.13 km (neg. direction)
Fy= FsenФ = 7.20sen55° = 5.90 km (neg. direction)
- Displacement G: 2.80 km and 10° north of east
Gx= GcosФ = 2.80cos10° = 2.76 km (pos. direction)
Gy= GsenФ = 2.80sen10° = 0.49 km (pos. direction)
Now, we add the components along the x- and y- axis to find the components of the resultant (R):
Rx = Ax + Bx + Cx + Dx + Ex + Fx + Gx
Ry = Ay + By + Cy + Dy + Ey + Fy + Gy
Therefore,
Rx= (-1.77)+(2.35)+(-1.18)+(5.10)+(0.15)+(-4.13)+(2.76)
Ry= (1.77)+(-4.07)+(-0.55)+(0)+(1.69)+(-5.90)+(0.49)
<em>Note: minus (-) symbol to negative directions</em>
Rx = 3.28 km
Ry = - 6.57 km
Let´s use the Theorem of Pythagoras to find the magnitud ot the resultant:
R = 7,34 km
To find the direction of the resultant:
tanФ =
tanФ =
tanФ = 2.00
Ф = tan-1 (2.00)
Ф = 63,43° north of east
