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, we need to find the net distance and direction traveled by the car. We can break down the distances traveled into their components to calculate the displacement.
Step 1: Draw a diagram to visualize the directions and distances traveled.
Start by drawing a coordinate system with the point of origin as the reference point. Since the car is initially driven 135 km west from the origin, mark a point to the left of the origin 135 units away. Next, the car travels 40 km southwest. Draw a line at a 45-degree angle to the southwest from the end point of the previous line.
Step 2: Calculate the horizontal and vertical components.
The 135 km traveled west is entirely horizontal, so the horizontal component is -135 km (negative because it is towards the left). The 40 km southwest can be broken down into horizontal and vertical components. To find the horizontal component, multiply the distance by the cosine of the angle (45 degrees), which gives us -40 km * cos(45) = -28.28 km. To find the vertical component, multiply the distance by the sine of the angle, which gives us -40 km * sin(45) = -28.28 km.
Step 3: Calculate the net horizontal and vertical components.
To find the net horizontal displacement, add the horizontal components of both distances: -135 km + (-28.28 km) = -135 km - 28.28 km = -163.28 km (negative because it is towards the left). To find the net vertical displacement, add the vertical components of both distances: 0 km + (-28.28 km) = -28.28 km (negative because it is downwards).
Step 4: Calculate the magnitude and direction of the displacement.
To find the magnitude of the displacement, use the Pythagorean theorem: magnitude = sqrt((vertical displacement)^2 + (horizontal displacement)^2) = sqrt((-28.28 km)^2 + (-163.28 km)^2) = sqrt(8007.78 km^2) ≈ 89.49 km.
To find the direction, use the inverse tangent function: direction = atan(vertical displacement / horizontal displacement) = atan(-28.28 km / -163.28 km) = atan(0.1733) ≈ 9.90 degrees.
So, the magnitude of the displacement is approximately 89.49 km, and the direction is approximately 9.90 degrees (west of south).