When two molecules of hydrogen combine with one molecule of oxygen to form two water molecules,

$$2 \mathrm{H}_2+\mathrm{O}_2 \longrightarrow 2 \mathrm{H}_2 \mathrm{O}$$

the energy released is $5 \mathrm{eV}$. (a) What is the mass difference between the three original molecules and the two final ones? (b) Given that the water molecule has mass about $3 \times 10^{-26} \mathrm{~kg}$, what is the fractional change in mass, $\Delta M /$ (total mass)? (Does it matter significantly whether you use the initial or final total mass?) (c) If one were to form $10 \mathrm{~g}$ of water by this process, what would be the total change in rest mass?

The rest energy of a certain nuclear particle is $5 \mathrm{GeV}\left(1 \mathrm{GeV}=10^9 \mathrm{eV}\right)$, and its kinetic energy is found to be $8 \mathrm{GeV}$. Find its momentum (in $\mathrm{GeV} / c$ ), and what is its speed?

At the Stanford Linear Accelerator, electrons are accelerated to energies of $50 \mathrm{GeV}\left(1 \mathrm{GeV}=10^9 \mathrm{eV}\right)$. (a) If this energy were classical kinetic energy, what would be the electrons' speed? (Take the electrons' mass to be $0.5 \mathrm{MeV} / \mathrm{c}^2$.) (b) Calculate $\gamma$ and hence Calculate the electrons' actual speed.

Find the momentum of a $1000-\mathrm{kg}$ space probe when traveling at the speed needed to escape the earth's gravitational attraction (about $10,000 \mathrm{~m} / \mathrm{s}$ )? What is the percent error from using the nonrelativistic formula $p=m v$ for momentum?

A proton (rest mass $938 \mathrm{MeV} / \mathrm{c}^2$ ) has kinetic energy $500 \mathrm{MeV}$. What is its momentum (in $\mathrm{MeV} / \mathrm{c}$ and in $\mathrm{kg} \cdot \mathrm{m} / \mathrm{s})$ ? How fast is it traveling?

A particle is observed with momentum $500 \mathrm{MeV} / c$ and energy $1746 \mathrm{MeV}$. What is its speed? What is its mass (in $\mathrm{MeV} / \mathrm{c}^2$ and in $\mathrm{kg}$ )?

A nuclear particle has mass $3 \mathrm{GeV} / \mathrm{c}^2$ and momentum $4 \mathrm{GeV} / \mathrm{c}$. (a) Find its energy? (b) Find its speed? $\left(1 \mathrm{GeV}=10^9 \mathrm{eV}\right.$. $)$

By squaring the definitions of $\mathbf{p}$ and $E$, verify the "Pythagorean relation" $E^2=(p c)^2+\left(m c^2\right)^2$.

(a) Assume that a mass $m$ has momentum $\mathbf{p}$ and energy $E$, as measured in a frame $S$. Use the relations (2.60) and (2.61) (Problem 2.10) and the known transformation of $d \mathbf{r}$ and $d t$ to find the values of $\mathbf{p}^{\prime}$ and $E^{\prime}$ as measured in a second frame $S^{\prime}$ traveling with speed $v$ along $O x$. (Notice that apart from some factors of $c$, the quantities $\mathbf{p}$ and $E$ transform just like $\mathbf{r}$ and $t$. Remember that $d t_0$ has the same value for all observers.) (b) Use the results of part (a) to prove the following important result: If the total momentum and energy of a system are conserved as measured in one inertial frame $S$, the same is true in any other inertial frame $S^{\prime}$.

If one defines a variable mass $m_{\text {var }}=\gamma m$, the relativistic momentum $\gamma m \mathbf{u}$ becomes $m_{\mathrm{var}} \mathbf{u}$, which looks more like the classical definition. Show, however, that the relativistic kinetic energy $(\gamma-1) m c^2$ is not equal to $\frac{1}{2} m_{\mathrm{var}} u^2$.

Suppose a relativistic, elastic, head-on collision of two particles as in Example $2.3$ (Section 2.4), but suppose that the second particle is much heavier than the first $\left(m_1 \ll m_2\right)$. Using $(2.19)$, show that $u_3 \approx-u_1$. That is, the first particle bounces back with it's speed unchanged (just as in the nonrelativistic case).

We saw that the relativistic momentum $\mathbf{p}=\gamma m \mathbf{u}$ of a mass $m$ can also be expressed as

$$\mathrm{p}=m \frac{d \mathrm{r}}{d t_0}$$

where $d t_0$ denotes the proper time between two neighboring points on the body's path and has the same value for all observers. Show that the relativistic energy $E=\gamma m c^2$ can similarly be rewritten as

$$E=m c^2 \frac{d t}{d t_0}$$

The above relations make it easy to see how $\mathbf{p}$ and $E$ transform from one inertial frame to another.

The mass of an electron is about $9.11 \times 10^{-31} \mathrm{~kg}$. Draw a table showing an electron's momentum, both the correct relativistic momentum and the nonrelativistic form, at speeds with $\beta=0.1,0.5,0.9,0.99$.

Harlen Industries has a simple forecasting model: Take the actual demand for the same month last year and divide that by the number of fractional weeks in that month. This gives the average weekly demand for that month. This weekly average is used as the weekly forecast for the same month this year. This technique was used to forecast eight weeks for this year, which are shown below along with the actual demand that occurred. The following eight weeks show the forecast (based on last year) and the demand that actually occurred. Using the RSFE, compute the tracking signal.