Chapter 7, Problem 7.74QP

(a)

Interpretation Introduction

Interpretation: The velocity of the given objects is to be calculated.

Concept introduction: The displacement of an object with respect to time is known as velocity. Velocity shows inverse relation with the wavelength of an object. It is measured in meter per second.

To determine: The velocity of a given object.

Answer to Problem 7.74QP

Solution

The velocity of an electron is $\underset{\_}{9.7\times 10^{2}m/s}$.

Explanation of Solution

Explanation

Given

Mass of an electron is $9.10938\times {10}^{-28}\text{g}$.

The conversion of g to kg is done as,

$1\text{g}={10}^{-3}\text{kg}$

Therefore the conversion of $9.10938\times {10}^{-28}\text{g}$ to kg is done as,

$\begin{array}{c}9.10938\times {10}^{-28}\text{g}=9.10938\times {10}^{-28}\times {10}^{-3}\text{kg}\\ =9.10938\times {10}^{-31}\text{kg}\end{array}$

The wavelength of an electron is $750\text{nm}$.

The conversion of nm to m is done as,

$1\text{nm}={10}^{-9}\text{m}$

Therefore, the conversion of $750\text{nm}$ to m is done as,

$\begin{array}{c}750\text{nm}=750\times {10}^{-9}\text{m}\\ =7.5\times {10}^{-7}\text{m}\end{array}$

The wavelength of a moving particle is given by the equation which is represented as,

$\lambda =\frac{h}{mu}$

Where,

• $h$ is the Planck’s constant $\left(6.626\times {10}^{-34}\text{J}\text{.s}\right)$.
• $u$ is the velocity in $\text{m/s}$.
• $m$ is the mass of the particle in $\text{kg}$.
• $\lambda$ is the wavelength of a particle in nm.

Rearranging the above equation in terms of velocity $\left(u\right)$,

$u=\frac{h}{m\lambda }$

Also, $1\text{J}=1\text{kg}{\text{.m}}^{\text{2}}{\text{/s}}^{\text{2}}$

Substitute the values of $h$, $\lambda$ and $m$ in the above equation to calculate the velocity.

$\begin{array}{c}u=\frac{6.626\times {10}^{-34}\text{kg}{\text{.m}}^{\text{2}}{\text{/s}}^{\text{2}}\text{.s}}{\left(9.10938\times {10}^{-31}\text{kg}\right)\left(7.5\times {10}^{-7}\text{m}\right)}\\ =\frac{6.626\times {10}^{-34}\text{kg}{\text{.m}}^{\text{2}}\text{/s}}{6.832\times {10}^{-37}\text{kg}\text{.m}}\\ =969.8\\ \approx \underset{\_}{9.7\times 10^{2}m/s}\end{array}$

Hence, the velocity of an electron is $\underset{\_}{9.7\times 10^{2}m/s}$.

(b)

Interpretation Introduction

To determine: The velocity of a given object.

Answer to Problem 7.74QP

Solution

The velocity of a proton is $\underset{\_}{5.283\times 10^{-1}m/s}$.

Explanation of Solution

Explanation

Given

Mass of a proton is $1.67262\times {10}^{-24}\text{g}$.

The conversion of g to kg is done as,

$1\text{g}={10}^{-3}\text{kg}$

Therefore the conversion of $1.67262\times {10}^{-24}\text{g}$ to kg is done as,

$\begin{array}{c}1.67262\times {10}^{-24}\text{g}=1.67262\times {10}^{-24}\times {10}^{-3}\text{kg}\\ =1.67262\times {10}^{-27}\text{kg}\end{array}$

The wavelength of a proton is $750\text{nm}$.

The conversion of nm to m is done as,

$1\text{nm}={10}^{-9}\text{m}$

Therefore, the conversion of $750\text{nm}$ to m is done as,

$\begin{array}{c}750\text{nm}=750\times {10}^{-9}\text{m}\\ =7.5\times {10}^{-7}\text{m}\end{array}$

The velocity of a proton is calculated by using the equation,

$u=\frac{h}{m\lambda }$

Also, $1\text{J}=1\text{kg}{\text{.m}}^{\text{2}}{\text{/s}}^{\text{2}}$

Substitute the values of $h$, $\lambda$ and $m$ in the above equation to calculate the velocity.

$\begin{array}{c}u=\frac{6.626\times {10}^{-34}\text{kg}{\text{.m}}^{\text{2}}{\text{/s}}^{\text{2}}\text{.s}}{\left(1.67262\times {10}^{-27}\text{kg}\right)\left(7.5\times {10}^{-7}\text{m}\right)}\\ =\frac{6.626\times {10}^{-34}\text{kg}{\text{.m}}^{\text{2}}\text{/s}}{1.254\times {10}^{-33}\text{kg}\text{.m}}\\ =\underset{\_}{5.283\times 10^{-1}m/s}\end{array}$

Hence, the velocity of a proton is $\underset{\_}{5.283\times 10^{-1}m/s}$.

(c)

Interpretation Introduction

To determine: The velocity of a given object.

Answer to Problem 7.74QP

Solution

The velocity of a neutron is $\underset{\_}{5.275\times 10^{-1}m/s}$.

Explanation of Solution

Explanation

Given

Mass of a neutron is $1.67493\times {10}^{-24}\text{g}$.

The conversion of g to kg is done as,

$1\text{g}={10}^{-3}\text{kg}$

Therefore the conversion of $1.67493\times {10}^{-24}\text{g}$ to kg is done as,

$\begin{array}{c}1.67493\times {10}^{-24}\text{g}=1.67493\times {10}^{-24}\times {10}^{-3}\text{kg}\\ =1.67493\times {10}^{-27}\text{kg}\end{array}$

The wavelength of a neutron is $750\text{nm}$.

The conversion of nm to m is done as,

$1\text{nm}={10}^{-9}\text{m}$

Therefore, the conversion of $750\text{nm}$ to m is done as,

$\begin{array}{c}750\text{nm}=750\times {10}^{-9}\text{m}\\ =7.5\times {10}^{-7}\text{m}\end{array}$

The velocity of a neutron is calculated by using the equation,

$u=\frac{h}{m\lambda }$

Also, $1\text{J}=1\text{kg}{\text{.m}}^{\text{2}}{\text{/s}}^{\text{2}}$

Substitute the values of $h$, $\lambda$ and $m$ in the above equation to calculate the velocity.

$\begin{array}{c}u=\frac{6.626\times {10}^{-34}\text{kg}{\text{.m}}^{\text{2}}{\text{/s}}^{\text{2}}\text{.s}}{\left(1.67493\times {10}^{-27}\text{kg}\right)\left(7.5\times {10}^{-7}\text{m}\right)}\\ =\frac{6.626\times {10}^{-34}\text{kg}{\text{.m}}^{\text{2}}\text{/s}}{1.256\times {10}^{-33}\text{kg}\text{.m}}\\ =\underset{\_}{5.275\times 10^{-1}m/s}\end{array}$

Hence, the velocity of a neutron is $\underset{\_}{5.275\times 10^{-1}m/s}$.

(d)

Interpretation Introduction

To determine: The velocity of a given object.

Answer to Problem 7.74QP

Solution

The velocity of an alpha particle is $\underset{\_}{1.33\times 10^{-1}m/s}$.

Explanation of Solution

Explanation

Given

Mass of an alpha particle is $6.64\times {10}^{-24}\text{g}$.

The conversion of g to kg is done as,

$1\text{g}={10}^{-3}\text{kg}$

Therefore the conversion of $6.64\times {10}^{-24}\text{g}$ to kg is done as,

$\begin{array}{c}6.64\times {10}^{-24}\text{g}=6.64\times {10}^{-24}\times {10}^{-3}\text{kg}\\ =6.64\times {10}^{-27}\text{kg}\end{array}$

The wavelength of an alpha particle is $750\text{nm}$.

The conversion of nm to m is done as,

$1\text{nm}={10}^{-9}\text{m}$

Therefore, the conversion of $750\text{nm}$ to m is done as,

$\begin{array}{c}750\text{nm}=750\times {10}^{-9}\text{m}\\ =7.5\times {10}^{-7}\text{m}\end{array}$

The velocity of an alpha particle is calculated by using the equation,

$u=\frac{h}{m\lambda }$

Also, $1\text{J}=1\text{kg}{\text{.m}}^{\text{2}}{\text{/s}}^{\text{2}}$

Substitute the values of $h$, $\lambda$ and $m$ in the above equation to calculate the velocity.

$\begin{array}{c}u=\frac{6.626\times {10}^{-34}\text{kg}{\text{.m}}^{\text{2}}{\text{/s}}^{\text{2}}\text{.s}}{\left(6.64\times {10}^{-27}\text{kg}\right)\left(7.5\times {10}^{-7}\text{m}\right)}\\ =\frac{6.626\times {10}^{-34}\text{kg}{\text{.m}}^{\text{2}}\text{/s}}{4.98\times {10}^{-33}\text{kg}\text{.m}}\\ =\underset{\_}{1.33\times 10^{-1}m/s}\end{array}$

Hence, the velocity of an alpha particle is $\underset{\_}{1.33\times 10^{-1}m/s}$.

Conclusion

1. a. The velocity of an electron is $\underset{\_}{9.7\times 10^{2}m/s}$.
2. b. The velocity of a proton is $\underset{\_}{5.283\times 10^{-1}m/s}$.
3. c. The velocity of a neutron is $\underset{\_}{5.275\times 10^{-1}m/s}$.
4. d. The velocity of an alpha particle is $\underset{\_}{1.33\times 10^{-1}m/s}$