A car is driven 30km west and then 80 km southwest. What is the displacement of the car from the point of origin (magnitude and direction)?
If you arrive at point denoted by the vector p,
p = -30i + (-80/√2i - 80/√2j)
p = -30i + (-56.57i - 56.57j)
p = -86.57i - 56.57j
|p| = 103.41
tanθ = -56.57/-86.57 = 0.653
θ = 213.16°
To find the displacement of the car, we need to determine the vector sum of the two individual displacements.
First, let's determine the x-component and y-component of each displacement.
1. Displacement 1: 30 km west
- x-component: -30 km (since west implies moving in the negative x-direction)
- y-component: 0 km (since there is no movement in the y-direction)
2. Displacement 2: 80 km southwest
- To determine the x-component and y-component of this displacement, we can break it down into its components using a right-angled triangle.
- The magnitude of the displacement is the hypotenuse of the triangle, which is 80 km.
- The angle between the southwest direction and the x-axis is 45 degrees.
- Using trigonometry, we can find the components:
- x-component: 80 km * cos(45 degrees) = 56.57 km (positive since it's in the positive x-direction)
- y-component: 80 km * sin(45 degrees) = 56.57 km (negative since it's in the negative y-direction)
Now, we can find the resultant displacement by adding the x-components and y-components separately.
Resultant x-component = -30 km + 56.57 km = 26.57 km (positive, since the positive x-direction is towards the east)
Resultant y-component = 0 km - 56.57 km = -56.57 km (negative, since the negative y-direction is towards the south)
Finally, we can combine the x-component and y-component to find the magnitude and direction of the displacement using the Pythagorean theorem and trigonometry.
Magnitude of the displacement:
Magnitude = sqrt((26.57 km)^2 + (-56.57 km)^2) ≈ 62.91 km
Direction of the displacement:
Direction = arctan((-56.57 km) / (26.57 km)) ≈ -64.39 degrees
Therefore, the magnitude of the displacement is approximately 62.91 km, and the direction is approximately 64.39 degrees south of west.
To find the displacement of the car, we need to determine the straight-line distance between the point of origin and the final position of the car. Here's how we can do that:
1. Represent the movements of the car using vectors.
- The first movement, 30 km west, can be represented as a vector pointing in the opposite direction of the displacement, i.e., 30 km towards the east.
- The second movement, 80 km southwest, can be represented as a vector pointing in the southwest direction.
2. Convert the southwest vector into its east and south components.
- To find the east component, we need to calculate the cosine of the angle between the southwest direction and the east direction. In this case, the angle between the southwest direction and the east direction is 45 degrees because southwest is halfway between west (90 degrees) and south (180 degrees). So, the east component is 80 km * cos(45 degrees).
- To find the south component, we need to calculate the sine of the angle between the southwest direction and the south direction. Since southwest is halfway between west and south, the angle is also 45 degrees. Therefore, the south component is 80 km * sin(45 degrees).
3. Sum up the east and west components to find the total eastward displacement.
- The initial westward displacement is 30 km. The southwest displacement has an east component that we calculated in the previous step. So, the total eastward displacement is 30 km - east component.
4. Sum up the north and south components to find the total northward displacement.
- The initial northward displacement is 0 km. The southwest displacement has a south component that we calculated in step 2. So, the total northward displacement is 0 km + south component.
5. Find the magnitude and direction of the displacement.
- The magnitude of the displacement is the straight-line distance between the initial position and the final position of the car, which can be calculated using the formula: magnitude = sqrt((east displacement)^2 + (north displacement)^2).
- The direction of the displacement can be determined by finding the angle between the positive east direction and the displacement vector. The angle can be calculated using the arctan function: direction = arctan(north displacement / east displacement).
By following these steps, you should be able to find the displacement of the car from the point of origin (magnitude and direction).