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 described here. The numbers may be different.
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To determine the car's total displacement, we need to calculate both the magnitude and direction of the displacement.
(a) To find the magnitude of the car's total displacement, we can use Pythagoras' theorem. The displacement along the horizontal (east-west) direction is 43 km, and the displacement along the vertical (north-south) direction is 25 km. These displacements can be seen as the two sides of a right-angled triangle. Using Pythagoras' theorem, the magnitude of the displacement is given by:
Magnitude = √(43^2 + 25^2)
Calculating this, we find:
Magnitude = √(1849 + 625) = √(2474) ≈ 49.74 km
Therefore, the magnitude of the car's total displacement is approximately 49.74 km.
(b) To find the angle of the car's total displacement measured from its starting direction, we can use trigonometry. We are given that the car travels in a direction 26° east of north for a distance of 26 km. This creates a right-angled triangle where the horizontal displacement (east-west) is 26 km and the vertical displacement (north-south) is 25 km.
The angle we need to find is the angle opposite to the horizontal displacement. Using trigonometry, we can determine this angle as follows:
tan(angle) = (opposite / adjacent) = (25 / 26)
angle = arctan(25 / 26)
Calculating this, we find:
angle = arctan(0.9615) ≈ 43.42°
Therefore, the angle of the car's total displacement measured from its starting direction is approximately 43.42° from east.