Completed Pass.The football team at Enormous State University (ESU) uses vector displacements to record its plays, with the origin taken to be the position of the ball before the play starts. In a certain pass play, the receiver starts at $\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{-}\mathbf{5}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$where the units are yards, data-custom-editor="chemistry" $\hat{\mathbf{i}}$ is to the right, and data-custom-editor="chemistry" $\hat{\mathbf{j}}$is downfield. Subsequent displacements of the receiver are data-custom-editor="chemistry" $\mathbf{+}\mathbf{9}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}$(he is in motion before the snap), data-custom-editor="chemistry" $\mathbf{+}\mathbf{11}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$(breaks downfield), data-custom-editor="chemistry" $\mathbf{-}\mathbf{6}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{4}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$(zigs), and data-custom-editor="chemistry" $\mathbf{+}\mathbf{12}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{14}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$(zags). Meanwhile, the quarterback has dropped straight back to a position data-custom-editor="chemistry" $\mathbf{-}\mathbf{7}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$. How far and in which direction must the quarterback throw the ball? (Like the coach, you will be well advised to diagram the situation before solving this numerically.)

Step by step solution

01

Given data

The receiver starts at $+1.0\hat{\mathrm{i}}-5.0\hat{\mathrm{j}}$,

Subsequent displacements of the receiver are as given below.

When he is in motion before the snap,$9\hat{\mathrm{i}}$

Breaks downfield,$+11.0\hat{\mathrm{j}}$

Zigs, $-6.0\hat{\mathrm{i}}+4.0\hat{\mathrm{j}}$

Zags, $+12.0\hat{\mathrm{i}}+18.0\hat{\mathrm{j}}$

The quarterback has dropped straight back to a position $-7.0\hat{\mathrm{j}}$.

02

Introduction

Unit vectors are the vectors whose magnitude is exactly 1 unit. They are very useful for various reasons. Specifically, the unit vectors [0,1]and [1,0] can together form any other vector.

03

Solution

Define the resultant vector as below.

$\begin{array}{rcl}\overrightarrow{\mathrm{R}}& =& \left(+1.0\hat{\mathrm{i}}-5.0\hat{\mathrm{j}}\right)+\left(+9.0\hat{\mathrm{i}}+11.0\hat{\mathrm{j}}\right)+\left(-6.0\hat{\mathrm{i}}+4.0\hat{\mathrm{j}}\right)+\left(+12\hat{\mathrm{i}}+18\hat{\mathrm{j}}\right)\\ & =& \left(1+9-6+12\right)\hat{\mathrm{i}}+\left(-5+11+4+18\right)\hat{\mathrm{j}}\\ & =& 16\hat{\mathrm{i}}+28\hat{\mathrm{j}}\end{array}$

Quarterback’s position $\overrightarrow{\mathrm{Q}}$is given by,

$\overrightarrow{\mathrm{Q}}=-7.0\hat{j}$

The relative distance between the two is given by,

$\begin{array}{rcl}\stackrel{\rightharpoonup }{\mathrm{R}}-\stackrel{\rightharpoonup }{\mathrm{Q}}& =& \left(16\hat{\mathrm{i}}+28\hat{\mathrm{j}}\right)-\left(-7.0\hat{\mathrm{j}}\right)\\ & =& 16.0\hat{\mathrm{i}}+35.0\hat{\mathrm{j}}\end{array}$

The direction is define as below.

(Both x and y components are positive)

$\begin{array}{rcl}\mathrm{\theta }& =& {\tan }^{-1}\left(\frac{35}{16}\right)\\ & =& {\tan }^{-1}\left(2.1875\right)\\ & =& 65.{43}^{\mathrm{o}}\end{array}$

Hence, the earthquake will throw the ball $38.48\mathrm{yards}$in one direction, with respect to $65.{43}^{\mathrm{o}}$.

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