a cross country runner runs 3km due east, 5km due south and 8km south west. what is the runners final displacement?
To find the runner's final displacement, we need to calculate both the horizontal and vertical components of the displacement.
First, let's break down the distances travelled into their respective components:
1. 3km due east is completely horizontal and has no vertical component.
2. 5km due south is completely vertical and has no horizontal component.
3. 8km southwest has both horizontal and vertical components.
To calculate the horizontal and vertical components of the third distance (8km southwest), we can use trigonometry. In this case, we have a right triangle with sides of 8km (hypotenuse), the horizontal component, and the vertical component.
Since southwest is 45 degrees between southwest and due west, we can use the sine and cosine functions:
Horizontal Component = 8km * cos(45°)
Vertical Component = 8km * sin(45°)
Using the trigonometric identities cos(45°) = sin(45°) = √2 / 2, we find:
Horizontal Component = 8km * (√2 / 2) = 4√2 km
Vertical Component = 8km * (√2 / 2) = 4√2 km
Now, let's add up the horizontal and vertical components:
Horizontal Displacement = 3km + (4√2 km)
Vertical Displacement = -5km + (4√2 km)
The negative sign for the vertical displacement indicates that it is in the opposite direction from the positive y-axis.
Finally, we can calculate the runner's final displacement using the Pythagorean theorem:
Final Displacement = √[(Horizontal Displacement)^2 + (Vertical Displacement)^2]
By substituting the values, we get:
Final Displacement = √[(3km + 4√2 km)^2 + (-5km + 4√2 km)^2]
We can then simplify and calculate the final displacement.
To find the runner's final displacement, we can use vector addition.
Let's break down the runner's displacement into east (x-coordinate) and south (y-coordinate) components.
1. The runner runs 3 km due east. This means the x-coordinate component is +3 km.
2. The runner then runs 5 km due south. This means the y-coordinate component is -5 km.
3. Lastly, the runner runs 8 km south-west. To find the components, we can split it into two perpendicular directions:
- The runner runs 8 km south, so the y-coordinate component becomes -5 km + (-8 km).
- The runner runs 8 km west, so the x-coordinate component becomes +3 km + (-8 km).
Adding up the x and y coordinate components, we get:
x-coordinate component: 3 km + (-8 km) = -5 km
y-coordinate component: -5 km + (-8 km) = -13 km
Therefore, the runner's final displacement is (-5 km, -13 km).