Chapter 9, Problem 1CQ

Rank in order, from largest to smallest, the momenta p_1x through p_5x of the objects presented in Figure Q9.1. Explain.

Figure Q9.1

To determine

To rank: In order from largest to smallest momentum for five different objects.

Answer to Problem 1CQ

The momentum for five different objects from largest to smallest ranked is $\underset{\_}{p_{2x}>p_{1x}=p_{3x}=p_{5x}>p_{4x}}$.

Explanation of Solution

Given data:

Refer to Figure Q9.1 in the textbook for five different objects with their masses and velocities.

From the figure, mass of the object 1 $\left(m_1\right)$ is 20 g.

Velocity of the object 1 $\left(v_{1x}\right)$ is $1\frac{\text{m}}{\text{s}}$.

Mass of the object 2 $\left(m_2\right)$ is 20 g.

Velocity of the object 2 $\left(v_{2x}\right)$ is $2\frac{\text{m}}{\text{s}}$.

Mass of the object 3 $\left(m_3\right)$ is 10 g.

Velocity of the object 3 $\left(v_{3x}\right)$ is $2\frac{\text{m}}{\text{s}}$.

Mass of the object 4 $\left(m_4\right)$ is 10 g.

Velocity of the object 4 $\left(v_{4x}\right)$ is $1\frac{\text{m}}{\text{s}}$.

Mass of the object 5 $\left(m_5\right)$ is 200 g.

Velocity of the object 5 $\left(v_{5x}\right)$ is $0.1\frac{\text{m}}{\text{s}}$.

Formula used:

Write the expression for momentum of an object (in x-direction) in terms of its mass and velocity as follows.

$p_x=m{v}_x$ (1)

Here,

$m$ is the mass of the object, and

$v_x$ is the velocity of the object in x-direction.

Calculation of momentum for object 1 $p_{1x}$:

Modify the expression in equation (1) to calculate the momentum of object 1 as follows.

$p_{1x}=m_1{v}_{1x}$

Substitute 20 g for $m_1$, and $1\frac{\text{m}}{\text{s}}$ for $v_{1x}$,

$\begin{array}{c}{p}_{1x}=\left(20\text{g}\right)\left(1\frac{\text{m}}{\text{s}}\right)\\ =20\frac{\text{g}\cdot \text{m}}{\text{s}}\end{array}$

Calculation of momentum for object 2 $p_{2x}$:

Modify the expression in equation (1) to calculate the momentum of object 2 as follows.

$p_{2x}=m_2{v}_{2x}$

Substitute 20 g for $m_2$, and $2\frac{\text{m}}{\text{s}}$ for $v_{2x}$,

$\begin{array}{c}{p}_{2x}=\left(20\text{g}\right)\left(2\frac{\text{m}}{\text{s}}\right)\\ =40\frac{\text{g}\cdot \text{m}}{\text{s}}\end{array}$

Calculation of momentum for object 3 $p_{3x}$:

Modify the expression in equation (1) to calculate the momentum of object 3 as follows.

$p_{3x}=m_3{v}_{3x}$

Substitute 10 g for $m_3$, and $2\frac{\text{m}}{\text{s}}$ for $v_{3x}$,

$\begin{array}{c}{p}_{3x}=\left(10\text{g}\right)\left(2\frac{\text{m}}{\text{s}}\right)\\ =20\frac{\text{g}\cdot \text{m}}{\text{s}}\end{array}$

Calculation of momentum for object 4 $p_{4x}$:

Modify the expression in equation (1) to calculate the momentum of object 4 as follows.

$p_{4x}=m_4{v}_{4x}$

Substitute 10 g for $m_4$, and $1\frac{\text{m}}{\text{s}}$ for $v_{4x}$,

$\begin{array}{c}{p}_{4x}=\left(10\text{g}\right)\left(1\frac{\text{m}}{\text{s}}\right)\\ =10\frac{\text{g}\cdot \text{m}}{\text{s}}\end{array}$

Calculation of momentum for object 5 $p_{5x}$:

Modify the expression in equation (1) to calculate the momentum of object 5 as follows.

$p_{5x}=m_5{v}_{5x}$

Substitute 200 g for $m_5$, and $0.1\frac{\text{m}}{\text{s}}$ for $v_{5x}$,

$\begin{array}{c}{p}_{5x}=\left(200\text{g}\right)\left(0.1\frac{\text{m}}{\text{s}}\right)\\ =20\frac{\text{g}\cdot \text{m}}{\text{s}}\end{array}$

From the calculations of the momentum of each object, the ranks are given from largest to the smallest as follows.

$p_{2x}>p_{1x}=p_{3x}=p_{5x}>p_{4x}$

Conclusion:

Thus, the momentum for five different objects from largest to smallest ranked is $\underset{\_}{p_{2x}>p_{1x}=p_{3x}=p_{5x}>p_{4x}}$.