Question

(figure 1) shows a map of olivias trip to a coffee shop. she gets on her bike at loomis and then rides south 0.9mi to broadway. she turns east onto broadway, rides 0.8 mi to where broadway turns, and then continues another 1.6mi to the shop. figure 1 of 1 start loomis broadway 30° end part a what is the magnitude of the total displacement of her trip? express your answer in miles. d = mi submit previous answers request answer

Explanation:

Step1: Resolve displacements into components

Let the south - north direction be along the y - axis (negative y for south) and east - west direction be along the x - axis (positive x for east).
The first displacement $\vec{d}_1$: She rides south for $d_{1y}=- 0.9$ mi and east for $d_{1x}=0$ mi.
The second displacement $\vec{d}_2$: The length of the second - part of the journey is $d_2 = 1.6$ mi at an angle $\theta = 30^{\circ}$ below the x - axis.
The x - component of the second displacement is $d_{2x}=d_2\cos\theta=1.6\cos30^{\circ}=1.6\times\frac{\sqrt{3}}{2}=0.8\sqrt{3}$ mi.
The y - component of the second displacement is $d_{2y}=-d_2\sin\theta=-1.6\sin30^{\circ}=- 0.8$ mi.

Step2: Calculate the total x - component and y - component of the displacement

The total x - component of the displacement $D_x=d_{1x}+d_{2x}=0 + 0.8\sqrt{3}\text{ mi}\approx1.39$ mi.
The total y - component of the displacement $D_y=d_{1y}+d_{2y}=-0.9-0.8=-1.7$ mi.

Step3: Calculate the magnitude of the total displacement

The magnitude of the displacement vector $\vec{D}$ is given by $D=\sqrt{D_x^{2}+D_y^{2}}$.
$D=\sqrt{(0.8\sqrt{3})^{2}+(-1.7)^{2}}=\sqrt{1.92 + 2.89}=\sqrt{4.81}\approx2.2$ mi.

Answer:

$2.2$ mi