Chapter 3, Problem 1PQ

A velocity vector has a magnitude of 720 m/s. Two students draw arrows representing this vector. Clarisse chooses a scale such that 1 cm → 100 m/s.

a. What is the length of the arrow that Clarisse draws?

b. Francois’s arrow is half as long as Clarisse’s. What is Francois’s scale?

c. Is one student’s choice better than the other? If so, what makes it a better scale?

To determine

The length of the arrow drawn to represent velocity vector using Clarisse’s scale.

Answer to Problem 1PQ

The length of the arrow is $7.20\text{cm}$.

Explanation of Solution

The magnitude of the velocity vector is $720{\text{ms}}^{-1}$. In Clarisse’s scale $1\text{cm}\to 1\text{00}{\text{ms}}^{-1}$.

When drawing an arrow to represent a vector, the length of the arrow is given by the magnitude of the vector. Thus, the length of the arrow should be $720{\text{ms}}^{-1}$.

Represent the length in Clarisse’s scale

$\begin{array}{c}720{\text{ ms}}^{-1}\to 720{\text{ ms}}^{-1}\cdot \frac{1\text{cm}}{100{\text{ms}}^{-1}}\\ \to 7.20\text{cm}\end{array}$

Conclusion:

Thus, the length of the arrow is $7.20\text{cm}$.

To determine

The scaling chosen by Francois’s scale.

Answer to Problem 1PQ

In Francois’s scale, $1\text{cm}\to 200{\text{ms}}^{-1}$.

Explanation of Solution

The length of Francois’s arrow is half of the length of Clarisse’s scale i.e. $\frac{7.20\text{}\text{cm}}{2}=3.60\text{cm}$. This length represents the magnitude of the vector i.e. $3.60\text{cm}\to \text{720}{\text{ms}}^{-1}$.

Divide on both sides by $3.60$

  $\begin{array}{c}1\text{cm}\to \frac{\text{720}{\text{ms}}^{-1}}{3.60}\\ \to 200{\text{ms}}^{-1}\end{array}$

Thus, in Francois’s scale, $1\text{cm}\to 200{\text{ms}}^{-1}$.

To determine

Explain whether one scale choice is better than the other.

Answer to Problem 1PQ

No, either of the scales can be used.

Explanation of Solution

Scaling is an individual’s preference and it doesn’t change the given values. As long as the scales accommodate the range of given values any scale can be used. The above given scales can be drawn to fit on a page and hence both of them are good choices.

Conclusion:

No, either of the scales can be used.