Chapter 20, Problem 6PQ

(a)

To determine

The average velocity of the particles.

Answer to Problem 6PQ

The average velocity of the particles is $\left(3.1\widehat{i}-1.1\widehat{j}-0.02\widehat{k}\right)\text{m/s}$.

Explanation of Solution

Write the equation for the average velocity in the x direction for the particles.

    ${\left(v_x\right)}_{av}=\frac{v_{1x}+v_{2x}+v_{3x}+v_{4x}+v_{5x}}{5}$                                                                     (I)

Here, ${\left(v_x\right)}_{av}$ is the average velocity in the x direction for the particles, $v_{1x}$ is the x component of the first particle, $v_{2x}$ is the x component of the second particle, $v_{3x}$ is the x component of the third particle, $v_{4x}$ is the x component of the fourth particle and $v_{5x}$ is the x component of the fifth particle.

Write the equation for the average velocity in the y direction for the particles.

    ${\left(v_y\right)}_{av}=\frac{v_{1y}+v_{2y}+v_{3y}+v_{4y}+v_{5y}}{5}$                                                                    (II)

Here, ${\left(v_y\right)}_{av}$ is the average velocity in the y direction for the particles, $v_{1y}$ is the y component of the first particle, $v_{2y}$ is the y component of the second particle, $v_{3y}$ is the y component of the third particle, $v_{4y}$ is the y component of the fourth particle and $v_{5y}$ is the y component of the fifth particle.

Write the equation for the average velocity in the z direction for the particles.

    ${\left(v_z\right)}_{av}=\frac{v_{1z}+v_{2z}+v_{3z}+v_{4z}+v_{5z}}{5}$                                                                     (III)

Here, ${\left(v_x\right)}_{av}$ is the average velocity in the z direction for the particles, $v_{1z}$ is the z component of the first particle, $v_{2z}$ is the z component of the second particle, $v_{3z}$ is the z component of the third particle, $v_{4z}$ is the z component of the fourth particle and $v_{5z}$ is the z component of the fifth particle.

Write the equation for the average velocity of the particles.

    ${\overrightarrow{v}}_{av}={\left(v_x\right)}_{av}\widehat{i}+{\left(v_y\right)}_{av}\widehat{j}+{\left(v_z\right)}_{av}\widehat{k}$                                                                  (IV)

Conclusion:

Substitute $3.4\text{m/s}$ for $v_{1x}$, $2.4\text{m/s}$ for $v_{2x}$, $-0.4\text{m/s}$ for $v_{3y}$, $9.2\text{m/s}$ for $v_{4y}$ and $0.9\text{m/s}$ for $v_{5y}$ in equation (I) to find ${\left(v_x\right)}_{av}$.

  $\begin{array}{c}{\left(v_x\right)}_{av}=\frac{3.4\text{m/s}+2.4\text{m/s}-0.4\text{m/s}+9.2\text{m/s}+0.9\text{m/s}}{5}\\ =3.1\text{m/s}\end{array}$

Substitute $5.6\text{m/s}$ for $v_{1y}$, $-6.6\text{m/s}$ for $v_{2y}$, $1.6\text{m/s}$ for $v_{3y}$, $-9.8\text{m/s}$ for $v_{4y}$ and $3.6\text{m/s}$ for $v_{5y}$ in equation (II) to find ${\left(v_y\right)}_{av}$.

  $\begin{array}{c}{\left(v_y\right)}_{av}=\frac{5.6\text{m/s}-6.6\text{m/s}+1.6\text{m/s}-9.8\text{m/s}+3.6\text{m/s}}{5}\\ =-1.1\text{m/s}\end{array}$

Substitute $6.7\text{m/s}$ for $v_{1z}$, $-3.7\text{m/s}$ for $v_{2z}$, $-10.7\text{m/s}$ for $v_{3z}$, $3.6\text{m/s}$ for $v_{4z}$ and $4.0\text{m/s}$ for $v_{5z}$ in equation (III) to find ${\left(v_z\right)}_{av}$.

  $\begin{array}{c}{\left(v_z\right)}_{av}=\frac{6.7\text{m/s}-3.7\text{m/s}-10.7\text{m/s}+3.6\text{m/s}+4.0\text{m/s}}{5}\\ =-0.02\text{m/s}\end{array}$

Substitute $3.1\text{m/s}$ for ${\left(v_x\right)}_{av}$, $-1.1\text{m/s}$ for ${\left(v_y\right)}_{av}$ and $-0.02\text{m/s}$ for ${\left(v_z\right)}_{av}$ in equation (IV) to find ${\overrightarrow{v}}_{av}$.

    $\begin{array}{c}{\overrightarrow{v}}_{av}=3.1\text{m/s}\widehat{i}-1.1\text{m/s}\widehat{j}-0.02\text{m/s}\widehat{k}\\ =\left(3.1\widehat{i}-1.1\widehat{j}-0.02\widehat{k}\right)\text{m/s}\end{array}$

Thus, the average velocity of the particles is $\left(3.1\widehat{i}-1.1\widehat{j}-0.02\widehat{k}\right)\text{m/s}$.

(b)

To determine

The average speed of the particles.

Answer to Problem 6PQ

The average speed of the particles is $9.5\text{m/s}$.

Explanation of Solution

Write the equation for the speedof the first particle.

    $v_1=\sqrt{{\left(v_{1x}\right)}^2+{\left(v_{1y}\right)}^2+{\left(v_{1z}\right)}^2}$                                                                  (V)

Write the equation for the average speed.

    $v_{av}=\frac{v_1+v_2+v_3+v_4+v_5}{5}$                                                                                (VI)

Conclusion:

Substitute $3.4\text{m/s}$ for $v_{1x}$, $5.6\text{m/s}$ for $v_{1y}$, $6.7\text{m/s}$ for $v_{1z}$ in equation (V) to find $v_1$.

  $\begin{array}{c}{v}_1=\sqrt{{\left(3.4\text{m/s}\right)}^2+{\left(5.6\text{m/s}\right)}^2+{\left(6.7\text{m/s}\right)}^2}\\ =9.4\text{m/s}\end{array}$

Similarly, calculate the speed of the second particle.

    $v_2=7.9\text{m/s}$

Calculate the speed of the third particle.

    $v_3=10.8\text{m/s}$

Calculate the speed of the fourth particle.

    $v_4=13.9\text{m/s}$

Calculate the speed of the fifth particle.

    $v_4=5.5\text{m/s}$

Substitute $9.4\text{m/s}$ for $v_1$, $7.9\text{m/s}$ for $v_2$, $10.8\text{m/s}$ for $v_3$, $13.9\text{m/s}$ for $v_4$ and $5.5\text{m/s}$ for $v_5$ in equation (VI) to find $v_{av}$.

    $\begin{array}{c}{v}_{av}=\frac{9.4\text{m/s}+7.9\text{m/s}+10.8\text{m/s}+13.9\text{m/s}+5.5\text{m/s}}{5}\\ =9.5\text{m/s}\end{array}$

Thus, the average speed of the particles is $9.5\text{m/s}$.

(c)

To determine

The rms speed of the particles.

Answer to Problem 6PQ

The rms speed of the particles is $9.9\text{m/s}$.

Explanation of Solution

Write the equation for the rms speed of the particles.

    $v_{rms}=\sqrt{\frac{1}{N}{\displaystyle \sum _{j=1}^N{v}_j^2}}$                                                                               (VII)

Conclusion:

Substitute the values for the speeds of the particles in equation (VII) to find $v_{rms}$.

  $\begin{array}{c}{v}_{rms}=\sqrt{\frac{1}{5}\left[{\left(9.4\text{m/s}\right)}^2+{\left(7.9\text{m/s}\right)}^2+{\left(10.8\text{m/s}\right)}^2+{\left(13.9\text{m/s}\right)}^2+{\left(5.5\text{m/s}\right)}^2\right]}\\ =9.9\text{m/s}\end{array}$

Thus, the rms speed of the particles is $9.9\text{m/s}$.

(d)

To determine

Compare the speeds of the particles.

Answer to Problem 6PQ

The rms speed is greater than the average speed.

Explanation of Solution

Write the equation for the magntide of the average velocity.

    $\left|{\overrightarrow{v}}_{avg}\right|=\sqrt{{\left(v_{av}\right)}_x^2+{\left(v_{av}\right)}_y^2+{\left(v_{av}\right)}_z^2}$                                                             (VIII)

Here, $\left|{\overrightarrow{v}}_{avg}\right|$ is the magntide of the average velocity

Conclusion:

Susbtitute $3.1\text{m/s}$ for ${\left(v_{av}\right)}_x$, $1.1\text{m/s}$ for ${\left(v_{av}\right)}_y$ and $0.20\text{m/s}$ for ${\left(v_{av}\right)}_y$ to find $\left|{\overrightarrow{v}}_{avg}\right|$.

  $\begin{array}{c}\left|{\overrightarrow{v}}_{avg}\right|=\sqrt{{\left(3.1\text{m/s}\right)}^2+{\left(1.1\text{m/s}\right)}^2+{\left(0.20\text{m/s}\right)}^2}\\ =3.3\text{m/s}\end{array}$

The value of rms speed is greater than the mahnitude of the average velocity. This is becauses, in calculationg the average velocity, the particles moving in the opposite direction is also considered.

Thus, the rms speed is greater than the average speed.