Questions: Jenny runs 1 mi to the northeast, then 1 mi south. Graphically find her net displacement.

Solution Steps

To find Jenny's net displacement, we can represent her path on a coordinate system. Assume she starts at the origin \((0, 0)\).

Running 1 mile to the northeast can be decomposed into its x and y components. Since northeast is at a 45-degree angle:

• \( x \)-component: \( 1 \cos(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071 \)
• \( y \)-component: \( 1 \sin(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071 \)

So, after the first leg of her run, her position is \((0.7071, 0.7071)\).

Next, Jenny runs 1 mile south. This affects only the y-component:

• New \( y \)-coordinate: \( 0.7071 - 1 = -0.2929 \)

Her new position is \((0.7071, -0.2929)\).

The net displacement is the straight-line distance from the origin to her final position. Using the distance formula: \[ \text{Displacement} = \sqrt{(0.7071 - 0)^2 + (-0.2929 - 0)^2} = \sqrt{0.7071^2 + (-0.2929)^2} \] \[ = \sqrt{0.5000 + 0.0858} = \sqrt{0.5858} \approx 0.7654 \text{ miles} \]

Final Answer