Question

22. II Susan, driving north at$60\text{\hspace{0.05em}\hspace{0.05em}mph}$ , and Trent, driving east at $45\text{mph}$ are approaching an intersection. What is Trent's speed relative to Susan's reference frame?

Short answer

The Trent's speed relative to Susan's reference frame is$75\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}mph}$.

Step-by-step solution

Step 1. Given information

Susan is driving north at $60\text{\hspace{0.05em}\hspace{0.05em}mph}$and Trent is driving east at $45\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}mph}$, and they are approaching an intersection.

The following figure shows the vector representation of the speed of the Susan with respect to Trent's reference.

Step 2. Explanation

The speed of the Trent with respect to the speed of the Susan's reference frame, from the shown figure (I) is,

${\left({\overrightarrow{V}}_{TS}\right)}^2={\left({\overrightarrow{V}}_{SG}\right)}^2+{\left({\overrightarrow{V}}_{TG}\right)}^2$

Here, ${\overrightarrow{V}}_{TS}$ is the speed of the Trent with respect to the speed of the Susan's reference frame, ${\overrightarrow{V}}_{SG}$is the speed of the Susan with respect to the ground, and${\overrightarrow{V}}_{TG}$ is the speed of the Trent with respect to ground.

$\begin{array}{l}{\left({\overrightarrow{V}}_{TS}\right)}^2={\left(60\text{mph}\right)}^2+{\left(45\text{mph}\right)}^2\\ {\overrightarrow{V}}_{TS}=\sqrt{5625{\text{mph}}^2}\\ =75\text{mph}\end{array}$

Thus, the Trent's speed relative to Susan's reference frame is$75\text{\hspace{0.05em}\hspace{0.05em}\hspace{0.05em}mph}$