A runner jobs 500 meters due was and then goes 300 meters due northwest. What is the runner's displacement?
To find the runner's displacement, we need to determine the straight line distance and direction from the starting point to the ending point. Let's break down the runner's movements and calculate the displacement:
1. The runner starts by jogging 500 meters due west.
- We can represent this movement as a vector with magnitude 500 meters in the west direction.
2. Next, the runner goes 300 meters due northwest.
- We can represent this movement as a vector with magnitude 300 meters in the northwest direction.
To calculate the runner's displacement, we need to find the vector sum of these two movements. Since these vectors are not in the same direction, we need to add them using vector addition.
To perform vector addition, we can break down each vector into its horizontal (east-west) and vertical (north-south) components. We can then add the components separately.
Let's calculate the horizontal and vertical components for each movement:
1. Jogging 500 meters due west:
- Horizontal component: -500 meters (negative because it's facing west)
- Vertical component: 0 meters (no north or south movement)
2. Going 300 meters due northwest:
- Horizontal component: 300 meters * cos(45 degrees) ≈ 212.12 meters (positive because it's facing northwest)
- Vertical component: 300 meters * sin(45 degrees) ≈ 212.12 meters (positive because it's facing northwest)
Now let's add the horizontal and vertical components separately:
Horizontal component: -500 meters + 212.12 meters ≈ -287.88 meters
Vertical component: 0 meters + 212.12 meters ≈ 212.12 meters
To get the magnitude (straight line distance) of the displacement, we can use the Pythagorean theorem:
Displacement = √((horizontal component)² + (vertical component)²)
Displacement = √((-287.88 meters)² + (212.12 meters)²)
Displacement ≈ √(82834.90 meters² + 44954.94 meters²)
Displacement ≈ √(127789.84 meters²)
Displacement ≈ 357.48 meters
Thus, the runner's displacement is approximately 357.48 meters in the northwest direction.