A car is driven 105 km west and then 65 km southwest. What is the displacement of the car from the point of origin (magnitude and direction)?
To find the displacement of a car from its point of origin, we need to calculate the net displacement vector.
First, let's break down the car's motion into its components. The car is driven 105 km west, so its displacement in the west direction (x-axis) would be -105 km.
Next, the car is driven 65 km southwest. To determine the displacement in this direction, we need to consider its components. The southwest direction is a combination of the south direction (y-axis) and the west direction (x-axis). Since it is not specified, we can assume the x-axis points to the right, and the y-axis points upwards.
We can calculate the displacement in the south direction by multiplying the distance (65 km) by the sine of the angle between the southwest direction and the south axis. In this case, the angle is 45 degrees, so the displacement in the south direction is 65 km * sin(45°) ≈ 45.91 km.
Similarly, we can calculate the displacement in the west direction by multiplying the distance (65 km) by the cosine of the angle between the southwest direction and the west axis. Again, the angle is 45 degrees, so the displacement in the west direction is 65 km * cos(45°) ≈ 45.91 km.
Now, we can add up the displacements in the x and y directions separately:
Displacement in the x-direction (west) = -105 km
Displacement in the y-direction (southwest) = -45.91 km + (-45.91 km) = -91.82 km
To find the net displacement, we can use these x and y components to calculate the magnitude and direction.
Magnitude of displacement = √((-105 km)^2 + (-91.82 km)^2) ≈ 138.24 km
To find the direction, we can use the inverse tangent function:
Direction = atan(-91.82 km / -105 km) ≈ 41.57°
Therefore, the magnitude of the displacement is approximately 138.24 km in the direction of about 41.57° relative to the positive x-axis.