A car is driven 125 km west and then 85 km southwest. What is the displacement of the car from the point of origin (magnitude and direction)?
find magnitude in km
find direction in degrees. South of west.
To find the displacement of the car, we can use the concept of vector addition.
Step 1: Draw a coordinate system with the origin as the starting point.
Step 2: Determine the vector representing the first leg of the journey, where the car travels 125 km west. This vector will be directed along the negative x-axis, since it is going west, and will have a magnitude of 125 km.
Step 3: Determine the vector representing the second leg of the journey, where the car travels 85 km southwest. This vector will have two components: one going southwest and another going west. To find these components, we can use basic trigonometry. The southwest component is the hypotenuse of a right triangle with a side length of 85 km, and we can use the angle provided to find the westward component.
Step 4: Add the two vectors together to find the resultant vector. This can be done by adding their x-components and their y-components separately.
Step 5: Calculate the magnitude of the resultant vector by using the Pythagorean theorem.
Step 6: Calculate the direction of the resultant vector using trigonometry. Since the resultant vector is in the third quadrant (southwest of the origin), we can find the angle by taking the inverse tangent of the ratio of the y-component to the x-component of the resultant vector, and then adding 180 degrees to account for the southwest direction.
Now, let's solve the problem:
Step 1: Draw a coordinate system with the origin as the starting point.
Y
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-------------------------> X
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Step 2: The first leg of the journey is 125 km west, so the vector representing this is (-125, 0).
Step 3: To determine the vector for the second leg, we need to find the southwest and west components. Let's use the given angle of southwest direction, which is typically 45 degrees.
The southwest component can be found using the Pythagorean theorem:
SW_component = 85 km * cos(45) ≈ 60.1 km
The westward component can be found using trigonometry:
West_component = 85 km * sin(45) ≈ 60.1 km
So, the vector for the second leg is (-60.1 km, -60.1 km).
Step 4: Adding the two vectors together:
Resultant vector = (-125 km, 0 km) + (-60.1 km, -60.1 km)
= (-185.1 km, -60.1 km)
Step 5: Calculating the magnitude of the resultant vector:
Magnitude = sqrt((-185.1 km)^2 + (-60.1 km)^2)
≈ 196.4 km
Step 6: Calculating the direction of the resultant vector:
Direction = arctan((-60.1 km)/(-185.1 km)) + 180 degrees
≈ 159.1 degrees
Therefore, the displacement of the car from the point of origin is approximately 196.4 km in magnitude, and it is directed at an angle of approximately 159.1 degrees south of west.