A car is driven east for a distance of 43 km, then north for 25 km, and then in a direction 26° east of north for 26 km. Determine (a) the magnitude (in km) of the car's total displacement from its starting point and (b) the angle (from east) of the car's total displacement measured from its starting direction.
Use the method of any of the "related problems" below. All they did was change the numbers a bit.
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To determine the car's total displacement, we can break down each leg of the journey into its horizontal (east/west) and vertical (north/south) components.
Leg 1: East 43 km
The horizontal component is 43 km east (positive).
The vertical component is 0 km (no north/south movement).
Leg 2: North 25 km
The horizontal component is 0 km (no east/west movement).
The vertical component is 25 km north (positive).
Leg 3: 26° east of north for 26 km
To find the horizontal and vertical components, we can use trigonometry. The angle 26° east of north is equivalent to 64° north of east.
Horizontal component = cos(64°) * 26 km
Vertical component = sin(64°) * 26 km
Now, we can calculate the values:
Horizontal component ≈ 0.448 * 26 km ≈ 11.648 km (east)
Vertical component ≈ 0.894 * 26 km ≈ 23.244 km (north)
Now, we can calculate the total displacement:
Horizontal displacement = 43 km (from leg 1) + 11.648 km (from leg 3) ≈ 54.648 km (east)
Vertical displacement = 25 km (from leg 2) + 23.244 km (from leg 3) ≈ 48.244 km (north)
(a) The magnitude of the car's total displacement is the straight-line distance between the starting point and the final point. We can use the Pythagorean theorem to find it:
Magnitude of displacement ≈ √(54.648 km)^2 + (48.244 km)^2 ≈ √(2995.813 km^2 + 2329.35 km^2) ≈ √5325.163 km^2 ≈ 72.986 km
Therefore, the magnitude of the car's total displacement is approximately 72.986 km.
(b) The angle of the car's total displacement, measured from its starting direction, can be found by using trigonometry. We can use the inverse tangent function to find the angle:
Angle = arctan(48.244 km / 54.648 km) ≈ arctan(0.883) ≈ 42.939°
Therefore, the angle of the car's total displacement, measured from the east direction, is approximately 42.939°.