A car is driven 135 km west and then 40 km southwest. What is the displacement of the car from the point of origin (magnitude and direction)?
To find the displacement of the car from the point of origin, we need to determine both the magnitude and direction of the displacement.
First, let's analyze the given information:
- The car is driven 135 km west.
- Then, the car is driven 40 km southwest.
To determine the displacement, we can break it down into its components using vector addition.
1. Start by drawing a coordinate system. Assume the car's initial position is the origin (0,0).
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|----------------- west (x-axis)
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2. Move 135 km west. This means the car's position will be (-135,0) on the grid.
3. Next, move 40 km southwest. This is a diagonal movement, which you can resolve into its x-component and y-component.
The southwest direction is a combination of moving south (-y direction) and west (-x direction). Since the southwest angle is not provided, we can assume a right triangle with sides of lengths 40 km.
Using the Pythagorean theorem, we can determine the x and y components of the southwest movement.
x-component: 40 km * cos(45°) = 40 km / √2 ≈ 28.28 km (rounded to two decimal places)
y-component: 40 km * sin(45°) = 40 km / √2 ≈ 28.28 km (rounded to two decimal places)
4. Add the x-components and y-components separately to find the net x and y displacements.
Net x displacement = -135 km + (-28.28 km) = -163.28 km (rounded to two decimal places)
Net y displacement = 0 km + (-28.28 km) = -28.28 km (rounded to two decimal places)
Now, we can calculate the magnitude and direction of the displacement using the net x and y components.
Magnitude of displacement (D) = √(Net x displacement)^2 + (Net y displacement)^2
D = √((-163.28 km)^2 + (-28.28 km)^2) ≈ 165.00 km (rounded to two decimal places)
To find the direction, we can use trigonometry to find the angle with respect to the x-axis.
Angle (θ) = arctan((Net y displacement) / (Net x displacement))
θ ≈ arctan((-28.28 km) / (-163.28 km)) ≈ arctan(0.1732) ≈ 9.94° (rounded to two decimal places)
Therefore, the displacement of the car from the point of origin is approximately 165.00 km in the direction of 9.94° from the west (clockwise from the positive x-axis).