Chapter 9, Problem 54P

(a)

To determine

To draw: The graph of the relativistic kinetic energy and the classical kinetic energy, both as a function of speed for an object.

Answer to Problem 54P

The graph of the classical kinetic energy and the speed is given in figure I and the graph of the relativistic kinetic energy and the speed is given in figure II.

Explanation of Solution

Consider the mass of the object is $1.00\text{kg}$, the rest energy of the object is $m{c}^2=9.00\times {10}^{16}\text{J}$.

The formula for the classical kinetic energy is,

$\begin{array}{c}{K}_{\text{c}}=\frac{1}{2}m{u}^2\\ =\frac{1}{2}m{c}^2{\left(\frac{u}{c}\right)}^2\end{array}$

Here,

$m$ is the mass of the object.

$u$ is the speed of the object.

$c$ is the speed of light.

Substitute $9.00\times {10}^{16}\text{J}$ for $m{c}^2$ in above equation.

$\begin{array}{c}{K}_{\text{c}}=\frac{1}{2}\left(9.00\times {10}^{16}\text{J}\right){\left(\frac{u}{c}\right)}^2\\ =\left(4.50\times {10}^{16}\text{J}\right){\left(\frac{u}{c}\right)}^2\end{array}$ (1)

The table for the classical kinetic energy with respective values of the speed, by substitute the values of $\frac{u}{c}$ in equation (1) is given as,

$\frac{u}{c}$	$K_{\text{c}}\left(\text{J}\right)$
0.000	0.000
0.100	0.045E16
0.200	0.180E16
0.300	0.405E16
0.400	0.720E16
0.500	1.13E16
0.600	1.62E16
0.700	2.21E16
0.800	2.88E16
0.900	3.65E16
0.990	4.41E16

Table 1

The graph for classical kinetic energy as a function of speed for an object is given below.

Figure I

The formula for the relativistic kinetic energy is,

$K_{\text{r}}=\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)m{c}^2$

Substitute $9.00\times {10}^{16}\text{J}$ for $m{c}^2$ in above equation.

$K_{\text{r}}=\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)\left(9.00\times {10}^{16}\text{J}\right)$ (2)

The table for the relativistic kinetic energy with respective values of the speed, by substitute the values of $\frac{u}{c}$ in equation (2) is given as,

$\frac{u}{c}$	$K_{\text{r}}\left(\text{J}\right)$
0.000	0.000
0.100	0.0453E16
0.200	0.186E16
0.300	0.435E16
0.400	0.820E16
0.500	1.39E16
0.600	2.25E16
0.700	3.60E16
0.800	6.00E16
0.900	11.6E16
0.990	54.8E16

Table 2

The graph for relativistic kinetic energy as a function of speed for an object is given below.

Figure II

Conclusion:

Therefore, the graph of the classical kinetic energy and the speed is given in figure I and the graph of the relativistic kinetic energy and the speed is given in figure II.

(b)

To determine

The speed at which the classical kinetic energy underestimate the experimental value by $1\%$.

Answer to Problem 54P

The speed at which the classical kinetic energy underestimate the experimental value by $1\%$ is $0.115c$.

Explanation of Solution

As the classical kinetic energy is underestimate the experimental value by $1\%$. Then,

$K_{\text{c}}=0.990K_{\text{r}}$

Substitute $\frac{1}{2}m{c}^2{\left(\frac{u}{c}\right)}^2$ for $K_{\text{c}}$ and $\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)m{c}^2$ for $K_{\text{r}}$.

$\begin{array}{c}\frac{1}{2}m{c}^2{\left(\frac{u}{c}\right)}^2=0.990\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)m{c}^2\\ {\left(\frac{u}{c}\right)}^2=\left(1.98\right)\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)\\ u=0.115c\end{array}$

Conclusion:

Therefore, the speed at which the classical kinetic energy underestimate the experimental value by $1\%$ is $0.115c$.

(c)

To determine

The speed at which the classical kinetic energy underestimate the experimental value by $5\%$.

Answer to Problem 54P

The speed at which the classical kinetic energy underestimate the experimental value by $5\%$ is $0.257c$.

Explanation of Solution

As the classical kinetic energy is underestimate the experimental value by $5\%$. Then,

$K_{\text{c}}=0.950K_{\text{r}}$

Substitute $\frac{1}{2}m{c}^2{\left(\frac{u}{c}\right)}^2$ for $K_{\text{c}}$ and $\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)m{c}^2$ for $K_{\text{r}}$.

$\begin{array}{c}\frac{1}{2}m{c}^2{\left(\frac{u}{c}\right)}^2=0.950\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)m{c}^2\\ {\left(\frac{u}{c}\right)}^2=\left(1.90\right)\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)\\ u=0.257c\end{array}$

Conclusion:

Therefore, the speed at which the classical kinetic energy underestimate the experimental value by $5\%$ is $0.257c$.

(d)

To determine

The speed at which the classical kinetic energy underestimate the experimental value by $50\%$.

Answer to Problem 54P

The speed at which the classical kinetic energy underestimate the experimental value by $50\%$ is $0.786c$.

Explanation of Solution

As the classical kinetic energy is underestimate the experimental value by $50\%$. Then,

$K_{\text{c}}=0.500K_{\text{r}}$

Substitute $\frac{1}{2}m{c}^2{\left(\frac{u}{c}\right)}^2$ for $K_{\text{c}}$ and $\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)m{c}^2$ for $K_{\text{r}}$.

$\begin{array}{c}\frac{1}{2}m{c}^2{\left(\frac{u}{c}\right)}^2=0.500\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)m{c}^2\\ {\left(\frac{u}{c}\right)}^2=\left(\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}-1\right)\\ u=0.786c\end{array}$

Conclusion:

Therefore, the speed at which the classical kinetic energy underestimate the experimental value by $50\%$ is $0.786c$.