A car is driven 145 km west and then 65 km southwest. What is the displacement of the car from the point of origin (magnitude and direction)?
what is the degrees south of west is what I need to know
tan^-1 [(south vector component)/(west vector component)]
= tan^-1 [(65*.707)/(65*.707 + 145)]
= tan^-1(0.2407)
= 13.5 degrees
To find the displacement of the car, we can use the Pythagorean theorem to calculate the magnitude and trigonometry to determine the direction.
First, let's break down the car's motion into its west and southwest components.
The car is driven 145 km west. This means its horizontal displacement, or west component, is 145 km.
Next, the car is driven 65 km southwest. To find the southwest component, we need to find the horizontal and vertical components of this displacement.
The southwest displacement has a westward component and a southward component. Since it is a right-angled triangle, we can use trigonometry to find the magnitude of the southwest displacement.
The southward component can be found by multiplying the hypotenuse (65 km) by the sine of the angle between the hypotenuse and the south direction.
sin(angle) = opposite / hypotenuse
sin(angle) = southward component / 65 km
Therefore, the southward component is equal to 65 km × sin(angle).
The westward component can be found by multiplying the hypotenuse (65 km) by the cosine of the angle between the hypotenuse and the west direction.
cos(angle) = adjacent / hypotenuse
cos(angle) = westward component / 65 km
Therefore, the westward component is equal to 65 km × cos(angle).
Now that we have the west component from the first displacement (145 km) and the westward and southward components from the second displacement, we can find the total westward and southward components.
Total westward component = 145 km + westward component
Total southward component = southward component
Using the Pythagorean theorem, we can find the magnitude of the displacement.
magnitude = √(total westward component^2 + total southward component^2)
Now we can calculate the direction of the displacement. The direction can be found using the tangent of the angle.
tan(angle) = total southward component / total westward component
angle = arctan(total southward component / total westward component)
Therefore, the displacement of the car from the point of origin is the magnitude (calculated using the Pythagorean theorem) and the direction (calculated using the arctan function) of the displacement vectors.