taylor decides to search for hard evidence to support his "they never landed on the moon theory" so he gets into his cactus powered mobile and drives. he begins by driving 50.0km [N]. he then turns and drives 30.0k [N30E]. he comes his futile search by driving 25.0 km [w] and then 40.0 km [sw]. if taylors entire trip took 8.5 hours, find his displacement
East distance = E = 0 + 30 sin 30 - 25 - 40 sin 45
North distance = N = 50 + 30 cos 30 + 0 - 40 cos 45
distance from origin (displacement magnitude) = sqrt (E^2+N^2)
tan (angle north of east) = N/E
All angles are measured CW from +y-axis.
Disp. = 50i + 30[30o] - 25 + 40[225o]
X = 30*sin30 - 25 + 40*sin225 = 15 - 25 - 28.3 = -38.3.
Y = 50i + 30*cos30 + 40*cos225 = 50i + 26i - 28.3i = 52.3i
Disp.. = -38.3 + 52.3i = 64.8km[-36.2] = 64.8km[36.2o] W. of N. = 324o CW
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To find Taylor's displacement, we need to calculate the straight-line distance and direction from his starting point to his ending point.
We can break down Taylor's movements into North (N), East (E), West (W), and South (S) components:
1. He drives 50.0 km to the North (N).
2. He then turns and drives 30.0 km in the direction N30E.
To calculate the North and East components of this leg, we use trigonometry:
North component: 30.0 km * cos(30°) = 30.0 km * (√3/2) = 30.0 km * 0.866 = 25.98 km (rounded to two decimal places)
East component: 30.0 km * sin(30°) = 30.0 km * (1/2) = 15.0 km
3. Next, he drives 25.0 km due West (W).
4. Finally, he drives 40.0 km in the southwest (SW) direction.
To calculate the South and West components of this leg, we again use trigonometry:
South component: 40.0 km * cos(45°) = 40.0 km * (1/√2) = 40.0 km * 0.707 = 28.28 km (rounded to two decimal places)
West component: 40.0 km * sin(45°) = 40.0 km * (1/√2) = 40.0 km * 0.707 = 28.28 km (rounded to two decimal places)
Now, let's calculate the total North and total South components:
Total North: 50.00 km + 25.98 km = 75.98 km (rounded to two decimal places)
Total South: 28.28 km
Lastly, we will calculate the total West and total East components:
Total West: 25.0 km + 28.28 km = 53.28 km (rounded to two decimal places)
Total East: 15.0 km
To find the displacement, we use the Pythagorean theorem:
Displacement = √((Total North - Total South)² + (Total West - Total East)²)
Displacement = √((75.98 km - 28.28 km)² + (53.28 km - 15.0 km)²)
Displacement = √(47.70 km² + 38.28 km²)
Displacement ≈ √(2296.49 km²)
Displacement ≈ 47.93 km
Therefore, Taylor's displacement is approximately 47.93 km.