Question

1. kelsey swims 1.5km east, then bikes 40km north, and then runs 10km west. based on the properties that define vector quantities in two - dimensional motion, which choice gives the correct solution for the magnitude of kelseys displacement? clear all o c²=(40²)+(8.5²) o c²=(40²)+(10²)+(1.5²) o c²=(1.5²)+(40²) o c²=(40²)+(10²)

Explanation:

Step1: Define vector components

Let the east - west direction be the x - axis (east is positive x, west is negative x) and north - south direction be the y - axis (north is positive y). The east - ward displacement $x_1 = 15$ km, the north - ward displacement $y_1=40$ km and the west - ward displacement $x_2=- 10$ km. The net x - component of the displacement $x=x_1 + x_2=15-10 = 5$ km and the y - component of the displacement $y = 40$ km.

Step2: Use the Pythagorean theorem for magnitude

The magnitude of the displacement vector $d$ in two - dimensions is given by $d=\sqrt{x^{2}+y^{2}}$. Substituting $x = 5$ km and $y = 40$ km, we can also calculate it using the individual displacements directly. The magnitude of the displacement $d$ of the motion is given by the Pythagorean theorem for vector addition. If we consider the three displacements, the magnitude of the net displacement $d$ of the motion is calculated as $d=\sqrt{(15 - 10)^2+(40)^2}=\sqrt{(5)^2+(40)^2}$. In terms of the original displacements without calculating the net x - component first, we know that for a right - angled triangle formed by the net displacement vector, the magnitude $d$ of the displacement is given by $d=\sqrt{(15)^2+(40)^2+( - 10)^2}=\sqrt{(15)^2+(40)^2+(10)^2}$. The formula for the magnitude of the displacement vector when adding vectors in two - dimensions is $d=\sqrt{d_x^{2}+d_y^{2}}$, where $d_x$ and $d_y$ are the net x and y components of the displacement. The correct formula for the magnitude of the displacement $C$ (using the Pythagorean theorem for vector addition) is $C=\sqrt{(15)^2+(40)^2+(10)^2}$, which is equivalent to $C^{2}=(15)^{2}+(40)^{2}+(10)^{2}$.

Answer:

The correct option is the one with $C^{2}=(15)^{2}+(40)^{2}+(10)^{2}$