A car is driven 125km west and 65km southwest. What is the displacement of the car from the point of origin (magnitude and direction)?
break the sw vector into s, and w components (it is 45 deg between S,W). then add like vectors.
To find the displacement of the car, we need to calculate the net distance and direction from the point of origin. Displacement refers to how far and in which direction an object is from its starting point.
First, let's break down the problem:
The car is driven 125 km west, and then 65 km southwest.
To determine the magnitude of the displacement, we can use the Pythagorean theorem. The theorem states that the square of the hypotenuse (the longest side) of a right-angled triangle is equal to the sum of the squares of the other two sides.
In this case, we have a right-angled triangle, and the westward distance traveled is the base, and the southwest distance is the perpendicular side. The hypotenuse of the triangle represents the direct displacement.
Using the given distances, we can calculate the magnitude of the displacement using the following steps:
Step 1: Calculate the square of the westward distance:
125 km west * 125 km west = 15,625 km²
Step 2: Calculate the square of the southwest distance:
65 km southwest * 65 km southwest = 4,225 km²
Step 3: Add the squares of the two distances:
15,625 km² + 4,225 km² = 19,850 km²
Step 4: Take the square root of the sum to find the magnitude of the displacement:
√19,850 km² ≈ 140.8 km
So, the magnitude of the displacement is approximately 140.8 km.
To find the direction of the displacement, we can use trigonometry. Since the westward distance forms the base and the southwest distance forms the perpendicular side of the triangle, we can use inverse tangent (tan⁻¹) to find the angle.
Step 5: Calculate the angle using inverse tangent:
tan⁻¹(65 km southwest / 125 km west) ≈ 27.7°
Therefore, the car's displacement from the point of origin is approximately 140.8 km in a direction of 27.7° (measured counterclockwise from due east).
To find the displacement of the car, we can use vector addition.
Step 1: Convert the distances to vector form.
- The distance driven 125 km west can be represented as (-125, 0).
- The distance driven 65 km southwest can be represented as (-65*cos(45°), -65*sin(45°)).
Step 2: Add the vectors together to find the combined displacement.
- (-125, 0) + (-65*cos(45°), -65*sin(45°)) = (-125 - 65*cos(45°), -65*sin(45°))
Step 3: Calculate the magnitude of the displacement using the Pythagorean theorem.
- Magnitude = √((-125 - 65*cos(45°))^2 + (-65*sin(45°))^2)
Step 4: Calculate the angle of the displacement using trigonometry.
- Angle = arctan((-65*sin(45°)) / (-125 - 65*cos(45°)))
By calculating the magnitude and direction of the displacement, we can determine the final answer.