A car is driven 110 km west and then 25 km southwest. What is the displacement of the car from the point of origin (magnitude and direction)?
...km
... degrees south of west
NOONE KNOWS HOW TO DO THIS.
To find the displacement of the car, we need to consider the vector sum of the two displacements: 110 km west and 25 km southwest.
First, let's draw a diagram to visualize the situation. On the diagram, we can draw a vector representing the displacement of 110 km to the west, and another vector representing the displacement of 25 km southwest.
The displacement of 110 km to the west can be represented by a vector pointing directly left.
The displacement of 25 km southwest can be broken down into two components: one component to the west and another component to the south. The west component can be obtained by drawing a line parallel to the west direction from the southwest vector. The length of this west component can be calculated using trigonometry. Since it forms a right triangle with the southwest vector, we can use the sine and cosine functions to find the length.
To calculate the length of the west component, we can use the given angle southwest vector makes with the west direction (which is the angle formed by the southwest vector and a line perpendicular to the west direction). You mentioned "south of west" in degrees, so let's assume the angle is 45 degrees (common for southwest).
Using trigonometry, the length of the west component can be calculated as follows:
west component length = 25 km * cos(45°)
Now that we have the lengths of the west and south components, we can add them to the 110 km west vector to find the displacement of the car.
Let's calculate the magnitudes of the two components first:
west component length = 25 km * cos(45°) ≈ 17.68 km
south component length = 25 km * sin(45°) ≈ 17.68 km
Now, let's calculate the displacement magnitude:
displacement magnitude = √((110 km + west component length)^2 + south component length^2)
= √((110 km + 17.68 km)^2 + (17.68 km)^2)
≈ √(12778.44 km^2 + 313.15 km^2)
≈ √13091.59 km^2
≈ 114.37 km
The displacement magnitude of the car from the point of origin is approximately 114.37 km.
To determine the direction, we need to calculate the angle the displacement vector makes with the west direction.
Let's calculate the direction:
angle = atan(south component length / (110 km + west component length))
angle ≈ atan(17.68 km / 127.68 km)
≈ atan(0.1383)
The angle is approximately 4.501 degrees.
Therefore, the displacement of the car from the point of origin is approximately 114.37 km in magnitude and it is approximately 4.501 degrees south of west.