Two protons, each having a speed of $0.985 c$ in the laboratory, are moving toward each other. Determine $(b)$ the total momentum of the two protons in the laboratory,

Show analytically that a particle with momentum $p$ and energy $E$ has a speed given by

$$v=\frac{p c^2}{E}=\frac{p c}{\sqrt{m^2 c^2+p^2}} .$$

How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The rest masses of a proton (including an electron), a neutron, and neutral helium are, respectively, $1.00783 \mathrm{u}$, $1.00867 \mathrm{u}$, and $4.00260 \mathrm{u}$. (This energy difference is called the total binding energy of the ${ }_2^4 \mathrm{He}$ nucleus.)

Two atomic clocks are carefully synchronized. One remains in New York, and the other is loaded on an airliner that travels at an average speed of 250 m/s and then returns to New York. When the plane returns, the elapsed time on the clock that stayed behind is 4.00 h. By how much will the readings of the two clocks differ, and which clock will show the shorter elapsed time?

An unknown particle is measured to have a negative charge and a speed of $2.24 \times 10^8 \mathrm{~m} / \mathrm{s}$. Its momentum is determined to be $3.07 \times 10^{-22} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}$. Identify the particle by finding its mass.

The Sun radiates energy at a rate of about $4 \times 10^{26} \mathrm{~W}$. Estimate how long the Sun could last if it radiated constantly at this rate.

The Sun radiates energy at a rate of about $4 \times 10^{26} \mathrm{~W}$. How long does it take for the Sun to lose a mass equal to that of Earth?

The Sun radiates energy at a rate of about $4 \times 10^{26} \mathrm{~W}$. At what rate is the Sun's mass decreasing?

What principal advantage does an electron microscope have over an optical microscope?

If $E$ is the total energy of a particle with zero potential energy, show that $d E / d p=v$, where $p$ and $v$ are the momentum and velocity of the particle, respectively.

If E is the total energy of a particle with zero potential energy, show that dE/dp=v, where p and v are the momentum and velocity of the particle, respectively.

What minimum amount of electromagnetic energy is needed to produce an electron and a positron together? A positron is a particle with the same mass as an electron, but has the opposite charge. (Note that electric charge is conserved in this process.)

A young-looking woman astronaut has just arrived home from a long trip. She rushes up to an old gray-haired man and in the ensuing conversation refers to him as her son. How might this be possible ?

At what speed $v$ will the length of a $1.00~\mathrm{m}$ stick look $10.0 \%$ shorter $(90.0 \mathrm{~cm})$ ?