Chapter 14

A particle with mass \(m\) has speed \(c / 2\) relative to inertial frame S. The particle collides with an identical particle at rest relative to frame \(S .\) Relative to \(S,\) what is the speed of a frame \(S^{\prime}\) in which the total momentum of these particles is zero? This frame is called the center of momentum frame.

An elementary particle produced in a laboratory experiment travels \(0.230 \mathrm{~mm}\) through the lab at a relative speed of \(0.960 \mathrm{c}\) before it decays (becomes another particle). (a) What is the proper lifetime of the particle? (b) What is the distance the particle travels as measured from its rest frame?

When you cough, you expel air at high speed through the trachca and upper bronchi so that the air will remove cxcess mucus lining the pathway. You produce the high speed by this procedure: You brcathe in a large amount of air, trap it by closing the glottis (the narrow opening in the larynx ), increase the air pressure by contracting the lungs, partially collapse the trachca and upper bronchi to narrow the pathway, and then expel the air through the pathway by suddenly reopening the glottis. Assume that during the expulsion the volume flow rate is \(7.0 \times 10^{-7} \mathrm{~m}^{3} / \mathrm{s}\). What multiple of \(343 \mathrm{~m} / \mathrm{s}\) (the specd of sound \(v_{s}\) ) is the airspeed through the trachea if the trachea diameter (a) remains its normal value of \(14 \mathrm{~mm}\) and (b) contracts to \(5.2 \mathrm{~mm}\) ?

A tin can has a total volume of \(1200 \mathrm{~cm}^{3}\) and a mass of \(130 \mathrm{~g}\). How many grams of lead shot of density \(11.4 \mathrm{~g} / \mathrm{cm}^{3}\) could it carry without sinking in water?

How much energy is released in the explosion of a fission bomb containing \(3.0 \mathrm{~kg}\) of fissionable material? Assume that \(0.10 \%\) of the mass is converted to relcased energy. (b) What mass of TNT would have to explode to provide the same energy release? Assume that each mole of TNT liberates \(3.4 \mathrm{MJ}\) of energy on exploding. The molecular mass of TNT is \(0.227 \mathrm{~kg} / \mathrm{mol}\). (c) For the same mass of explosive, what is the ratio of the energy released in a nuclear explosion to that released in a TNI explosion?

The tension in a string holding a solid block below the surface of a liquid (of density greater than the block) is \(T_{0}\) when the container (Fig. \(14-57\) ) is at rest. When the container is given an upward acceleration of \(0.250 \mathrm{~g}\), what multiple of \(T_{0}\) gives the tension in the string?

What is the minimum area (in square meters) of the top surface of an ice slab \(0.441 \mathrm{~m}\) thick floating on fresh water that will hold up a \(938 \mathrm{~kg}\) automobile? Take the densities of ice and fresh water to be \(917 \mathrm{~kg} / \mathrm{m}^{3}\) and \(998 \mathrm{~kg} / \mathrm{m}^{3},\) respectively.

A Foron cruiser moving directly toward a Reptulian scout ship fires a decoy toward the scout ship. Relative to the scout ship, the speed of the decoy is \(0.980 \mathrm{c}\) and the speed of the Foron cruiser is \(0.900 \mathrm{c}\). What is the speed of the decoy relative to the cruiser?

A \(8.60 \mathrm{~kg}\) sphere of radius \(6.22 \mathrm{~cm}\) is at a depth of \(2.22 \mathrm{~km}\) in seawater that has an average density of \(1025 \mathrm{~kg} / \mathrm{m}^{3}\). What are the (a) gauge pressure, (b) total pressure, and (c) corrcsponding total force compressing the sphere's surface? What are (d) the magnitude of the buoyant force on the sphere and (c) the magnitude of the sphere's acceleration if it is free to move? Take atmospheric pressure to be \(1.01 \times 10^{5} \mathrm{~Pa}\).

For seawater of density \(1.03 \mathrm{~g} / \mathrm{cm}^{3}\), find the weight of water on top of a submarine at a depth of \(255 \mathrm{~m}\) if the horizontal crosssectional hull area is \(2200.0 \mathrm{~m}^{2}\). (b) In atmospheres, what water pressure would a diver experience at this depth?

Space cruisers \(A\) and \(B\) are moving parallel to the positive direction of an \(x\) axis. Cruiser \(A\) is faster, with a relative speed of \(v=0.900 c,\) and has a proper length of \(L=200 \mathrm{~m}\). According to the pilot of \(A\), at the instant \((t=0)\) the tails of the cruisers are aligned, the noses are also. According to the pilot of \(B,\) how much later are the noses aligned?

The scwage outlet of a house constructed on a slope is \(6.59 \mathrm{~m}\) below street level. If the sewer is \(2.16 \mathrm{~m}\) below street level, find the minimum pressure difference that must be created by the sewage pump to transfer waste of average density \(1000.00 \mathrm{~kg} / \mathrm{m}^{3}\) from outlet to sewer.

A relativistic train of proper length \(200 \mathrm{~m}\) approaches tunnel of the same proper length, at a relative speed of \(0.900 c\). A paint bomb in the engine room is set to explode (and cover everyone with blue paint) when the front of the train passes the far end of the tunnel (event FF). However, when the rear car passes the near end of the tunnel (event \(\mathrm{RN}\) ), a device in that car is set to send a signal to the engine room to deactivate the bomb. Train view: (a) What is the tunnel length? (b) Which event occurs first, FF or RN? (c) What is the time between those events? (d) Does the paint bomb explode? Tunnel view: (e) What is the train length? (f) Which event occurs first? (g) What is the time between those events? (h) Does the paint bomb explode? If your answers to (d) and (h) differ, you need to explain the paradox, because either the engine room is covered with blue paint or not; you cannot have it both ways. If your answers are the same, you need to explain why.

Ionization measurements show that a particular lightweight nuclear particle carries a double charge \((=2 e)\) and is moving with a speed of \(0.710 c\). Its measured radius of curvature in a magnetic field of \(1.00 \mathrm{~T}\) is \(6.28 \mathrm{~m}\). Find the mass of the particle and identify it. (Hints: Lightweight nuclear particles are made up of neutrons (which have no charge) and protons (charge \(=+e\) ), in roughly cqual numbers. Take the mass of cach such particle to be 1.00 u.) (See Problem 53.)

An astronaut exercising on a treadmill maintains a pulse rate of 150 per minute. If he exercises for \(1.00 \mathrm{~h}\) as measured by a clock on his spaceship, with a stride length of \(1.00 \mathrm{~m} / \mathrm{s},\) while the ship travels with a speed of \(0.900 \mathrm{c}\) relative to a ground station, what are (a) the pulse rate and (b) the distance walked as measured by someone at the ground station?

A spaceship approaches Earth at a speed of \(0.42 c .\) A light on the front of the ship appears red (wavelength \(650 \mathrm{nm}\) ) to passengers on the ship. What (a) wavelength and (b) color (blue, green, or yellow) would it appear to an observer on Earth?

Some of the familiar hydrogen lines appear in the spectrum of quasar \(3 \mathrm{C} 9,\) but they are shifted so far toward the red that their wavelengths are observed to be 3.0 times as long as those observed for hydrogen atoms at rest in the laboratory. (a) Show that the classical Doppler cquation gives a relative velocity of recession greater than \(c\) for this situation. (b) Assuming that the relative motion of \(3 \mathrm{C} 9\) and Earth is due entirely to the cosmological expansion of the universe, find the recession speed that is predicted by the relativistic Doppler equation.

Quite apart from effects due to Earth's rotational and orbital motions, a laboratory reference frame is not strictly an inertial frame because a particle at rest there will not, in general, remain at rest; it will fall. Often, however, events happen so quickly that we can ignore the gravitational acceleration and treat the frame as inertial. Consider, for example, an electron of speed \(v=0.992 c\) projected horizontally into a laboratory test chamber and moving through a distance of \(20 \mathrm{~cm}\). (a) How long would that take, and (b) how far would the electron fall during this interval? (c) What can you conclude about the suitability of the laboratory as an inertial frame in this case?

What is the speed parameter for the following speeds: (a) a typical rate of continental drift ( 1 in \(/ y\) ); (b) a typical drift speed for electrons in a current-carrying conductor \((0.5 \mathrm{~mm} / \mathrm{s}) ;\) (c) a highway speed limit of \(55 \mathrm{mi} / \mathrm{h} ;\) (d) the root-mean-square speed of a hydrogen molccule at room temperature; (c) a supersonic plane flying at Mach \(2.5(1200 \mathrm{~km} / \mathrm{h}) ;\) (f) the escape speed of a projectile from the Earth's surface; (g) the speed of Earth in its orbit around the Sun; (h) a typical recession speed of a distant quasar due to the cosmological expansion \(\left(3.0 \times 10^{4} \mathrm{~km} / \mathrm{s}\right) ?\)