Chapter 7

The nucleus of radioactive thorium- 228 , with a mass of about \(3.8 \cdot 10^{-25} \mathrm{~kg}\), is known to decay by emitting an alpha particle with a mass of about \(6.68 \cdot 10^{-27} \mathrm{~kg} .\) If the alpha particle is emitted with a speed of \(1.8 \cdot 10^{7} \mathrm{~m} / \mathrm{s},\) what is the recoil speed of the remaining nucleus (which is the nucleus of a radon atom)?

A 60.0 -kg astronaut inside a 7.00 -m-long space capsule of mass \(500 . \mathrm{kg}\) is floating weightlessly on one end of the capsule. He kicks off the wall at a velocity of \(3.50 \mathrm{~m} / \mathrm{s}\) toward the other end of the capsule. How long does it take the astronaut to reach the far wall?

Moessbauer spectroscopy is a technique for studying molecules by looking at a particular atom within them. For example, Moessbauer measurements of iron (Fe) inside hemoglobin, the molecule responsible for transporting oxygen in the blood, can be used to determine the hemoglobin's flexibility. The technique starts with X-rays emitted from the nuclei of \({ }^{57}\) Co atoms. These X-rays are then used to study the Fe in the hemoglobin. The energy and momentum of each X-ray are \(14 \mathrm{keV}\) and \(14 \mathrm{keV} / \mathrm{c}\) (see Example 7.5 for an explanation of the units). \(\mathrm{A}^{57}\) Co nucleus recoils as an \(\mathrm{X}\) -ray is emitted. A single \({ }^{57}\) Co nucleus has a mass of \(9.52 \cdot 10^{-26} \mathrm{~kg} .\) What are the final momentum and kinetic energy of the \({ }^{57}\) Co nucleus? How do these compare to the values for the X-ray?

Assume the nucleus of a radon atom, \({ }^{222} \mathrm{Rn}\), has a mass of \(3.68 \cdot 10^{-25} \mathrm{~kg} .\) This radioactive nucleus decays by emitting an alpha particle with an energy of \(8.79 \cdot 10^{-13} \mathrm{~J}\). The mass of an alpha particle is \(6.65 \cdot 10^{-27} \mathrm{~kg}\). Assuming that the radon nucleus was initially at rest, what is the velocity of the nucleus that remains after the decay?

A skateboarder of mass \(35.0 \mathrm{~kg}\) is riding her \(3.50-\mathrm{kg}\) skateboard at a speed of \(5.00 \mathrm{~m} / \mathrm{s}\). She jumps backward off her skateboard, sending the skateboard forward at a speed of \(8.50 \mathrm{~m} / \mathrm{s}\). At what speed is the skateboarder moving when her feet hit the ground?

During an ice-skating extravaganza, Robin Hood on Ice, a 50.0 -kg archer is standing still on ice skates. Assume that the friction between the ice skates and the ice is negligible. The archer shoots a \(0.100-\mathrm{kg}\) arrow horizontally at a speed of \(95.0 \mathrm{~m} / \mathrm{s} .\) At what speed does the archer recoil?

Astronauts are playing catch on the International Space Station. One 55.0 -kg astronaut, initially at rest, throws a baseball of mass \(0.145 \mathrm{~kg}\) at a speed of \(31.3 \mathrm{~m} / \mathrm{s}\). At what speed does the astronaut recoil?

A bungee jumper with mass \(55.0 \mathrm{~kg}\) reaches a speed of \(13.3 \mathrm{~m} / \mathrm{s}\) moving straight down when the elastic cord tied to her feet starts pulling her back up. After \(0.0250 \mathrm{~s},\) the jumper is heading back up at a speed of \(10.5 \mathrm{~m} / \mathrm{s}\). What is the average force that the bungee cord exerts on the jumper? What is the average number of \(g\) 's that the jumper experiences during this direction change?

A 3.0 -kg ball of clay with a speed of \(21 \mathrm{~m} / \mathrm{s}\) is thrown against a wall and sticks to the wall. What is the magnitude of the impulse exerted on the ball?

Tennis champion Venus Williams is capable of serving a tennis ball at around 127 mph. a) Assuming that her racquet is in contact with the 57.0 -g ball for \(0.250 \mathrm{~s}\), what is the average force of the racquet on the ball? b) What average force would an opponent's racquet have to exert in order to return Williams's serve at a speed of \(50.0 \mathrm{mph}\), assuming that the opponent's racquet is also in contact with the ball for 0.250 s?

Three birds are flying in a compact formation. The first bird, with a mass of \(100 . \mathrm{g}\) is flying \(35.0^{\circ}\) east of north at a speed of \(8.00 \mathrm{~m} / \mathrm{s}\). The second bird, with a mass of \(123 \mathrm{~g}\), is flying \(2.00^{\circ}\) east of north at a speed of \(11.0 \mathrm{~m} / \mathrm{s}\). The third bird, with a mass of \(112 \mathrm{~g}\), is flying \(22.0^{\circ}\) west of north at a speed of \(10.0 \mathrm{~m} / \mathrm{s}\). What is the momentum vector of the formation? What would be the speed and direction of a \(115-\mathrm{g}\) bird with the same momentum?

A golf ball of mass \(45.0 \mathrm{~g}\) moving at a speed of \(120 . \mathrm{km} / \mathrm{h}\) collides head on with a French TGV high-speed train of mass \(3.8 \cdot 10^{5} \mathrm{~kg}\) that is traveling at \(300 . \mathrm{km} / \mathrm{h}\). Assuming that the collision is elastic, what is the speed of the golf ball after the collision? (Do not try to conduct this experiment!)

In bocce, the object of the game is to get your balls (each with mass \(M=1.00 \mathrm{~kg}\) ) as close as possible to the small white ball (the pallina, mass \(m=0.045 \mathrm{~kg}\) ). Your first throw positioned your ball \(2.00 \mathrm{~m}\) to the left of the pallina. If your next throw has a speed of \(v=1.00 \mathrm{~m} / \mathrm{s}\) and the coefficient of kinetic friction is \(\mu_{\mathrm{k}}=0.20\), what are the final distances of your two balls from the pallina in each of the following cases? a) You throw your ball from the left, hitting your first ball. b) You throw your ball from the right, hitting the pallina.

A bored boy shoots a soft pellet from an air gun at a piece of cheese with mass \(0.25 \mathrm{~kg}\) that sits, keeping cool for dinner guests, on a block of ice. On one particular shot, his 1.2-g pellet gets stuck in the cheese, causing it to slide \(25 \mathrm{~cm}\) before coming to a stop. According to the package the gun came in, the muzzle velocity is \(65 \mathrm{~m} / \mathrm{s}\). What is the coefficient of friction between the cheese and the ice?

Some kids are playing a dangerous game with fireworks. They strap several firecrackers to a toy rocket and launch it into the air at an angle of \(60^{\circ}\) with respect to the ground. At the top of its trajectory, the contraption explodes, and the rocket breaks into two equal pieces. One of the pieces has half the speed that the rocket had before it exploded and travels straight upward with respect to the ground. Determine the speed and direction of the second piece.

A soccer ball with mass \(0.265 \mathrm{~kg}\) is initially at rest and is kicked at an angle of \(20.8^{\circ}\) with respect to the horizontal. The soccer ball travels a horizontal distance of \(52.8 \mathrm{~m}\) after it is kicked. What is the impulse received by the soccer ball during the kick? Assume there is no air resistance.

Tarzan, King of the Jungle (mass \(=70.4 \mathrm{~kg}\) ), grabs a vine of length \(14.5 \mathrm{~m}\) hanging from a tree branch. The angle of the vine was \(25.9^{\circ}\) with respect to the vertical when he grabbed it. At the lowest point of his trajectory, he picks up Jane (mass \(=43.4 \mathrm{~kg}\) ) and continues his swinging motion. What angle relative to the vertical will the vine have when Tarzan and Jane reach the highest point of their trajectory?

A bullet with mass \(35.5 \mathrm{~g}\) is shot horizontally from a gun. The bullet embeds in a 5.90 -kg block of wood that is suspended by strings. The combined mass swings upward, gaining a height of \(12.85 \mathrm{~cm}\). What was the speed of the bullet as it left the gun? (Air resistance can be ignored here.)

A 170.-g hockey puck moving in the positive \(x\) direction at \(30.0 \mathrm{~m} / \mathrm{s}\) is struck by a stick at time \(t=2.00 \mathrm{~s}\) and moves in the opposite direction at \(25.0 \mathrm{~m} / \mathrm{s}\). If the puck is in contact with the stick for \(0.200 \mathrm{~s}\), plot the momentum and the position of the puck, and the force acting on it as a function of time, from 0 to \(5.00 \mathrm{~s}\). Be sure to label the coordinate axes with reasonable numbers.

After several large firecrackers have been inserted into its holes, a bowling ball is projected into the air using a homemade launcher and explodes in midair. During the launch, the 7.00 -kg ball is shot into the air with an initial speed of \(10.0 \mathrm{~m} / \mathrm{s}\) at a \(40.0^{\circ}\) angle; it explodes at the peak of its trajectory, breaking into three pieces of equal mass. One piece travels straight up with a speed of \(3.00 \mathrm{~m} / \mathrm{s}\). Another piece travels straight back with a speed of \(2.00 \mathrm{~m} / \mathrm{s}\). What is the velocity of the third piece (speed and direction)?

In waterskiing, a "garage sale" occurs when a skier loses control and falls and waterskis fly in different directions. In one particular incident, a novice skier was skimming across the surface of the water at \(22.0 \mathrm{~m} / \mathrm{s}\) when he lost control. One ski, with a mass of \(1.50 \mathrm{~kg},\) flew off at an angle of \(12.0^{\circ}\) to the left of the initial direction of the skier with a speed of \(25.0 \mathrm{~m} / \mathrm{s}\). The other identical ski flew from the crash at an angle of \(5.00^{\circ}\) to the right with a speed of \(21.0 \mathrm{~m} / \mathrm{s} .\) What was the velocity of the \(61.0-\mathrm{kg}\) skier? Give a speed and a direction relative to the initial velocity vector.

An uncovered hopper car from a freight train rolls without friction or air resistance along a level track at a constant speed of \(6.70 \mathrm{~m} / \mathrm{s}\) in the positive \(x\) -direction. The mass of the car is \(1.18 \cdot 10^{5} \mathrm{~kg}\). a) As the car rolls, a monsoon rainstorm begins, and the car begins to collect water in its hopper (see the figure). What is the speed of the car after \(1.62 \cdot 10^{4} \mathrm{~kg}\) of water collects in the car's hopper? Assume that the rain is falling vertically in the negative \(y\) -direction. b) The rain stops, and a valve at the bottom of the hopper is opened to release the water. The speed of the car when the valve is opened is again \(6.70 \mathrm{~m} / \mathrm{s}\) in the positive \(x\) -direction (see the figure). The water drains out vertically in the negative \(y\) -direction. What is the speed of the car after all the water has drained out?

When a \(99.5-\mathrm{g}\) slice of bread is inserted into a toaster, the toaster's ejection spring is compressed by \(7.50 \mathrm{~cm}\). When the toaster ejects the toasted slice, the slice reaches a height \(3.0 \mathrm{~cm}\) above its starting position. What is the average force that the ejection spring exerts on the toast? What is the time over which the ejection spring pushes on the toast?

A potato cannon is used to launch a potato on a frozen lake, as shown in the figure. The mass of the cannon, \(m_{c}\) is \(10.0 \mathrm{~kg},\) and the mass of the potato, \(m_{\mathrm{p}}\), is \(0.850 \mathrm{~kg} .\) The cannon's spring (with spring constant \(\left.k_{\mathrm{c}}=7.06 \cdot 10^{3} \mathrm{~N} / \mathrm{m}\right)\) is compressed \(2.00 \mathrm{~m}\). Prior to launching the potato, the cannon is at rest. The potato leaves the cannon's muzzle moving horizontally to the right at a speed of \(v_{\mathrm{p}}=175 \mathrm{~m} / \mathrm{s}\). Neglect the effects of the potato spinning. Assume there is no friction between the cannon and the lake's ice or between the cannon barrel and the potato. a) What are the direction and magnitude of the cannon's velocity, \(\mathrm{v}_{c}\), after the potato leaves the muzzle? b) What is the total mechanical energy (potential and kinetic) of the potato/cannon system before and after the firing of the potato?

A particle \(\left(M_{1}=1.00 \mathrm{~kg}\right)\) moving at \(30.0^{\circ}\) downward from the horizontal with \(v_{1}=2.50 \mathrm{~m} / \mathrm{s}\) hits a second particle \(\left(M_{2}=2.00 \mathrm{~kg}\right),\) which was at rest momentarily. After the collision, the speed of \(M_{1}\) was reduced to \(.500 \mathrm{~m} / \mathrm{s}\), and it was moving at an angle of \(32^{\circ}\) downward with respect to the horizontal. Assume the collision is elastic. a) What is the speed of \(M_{2}\) after the collision? b) What is the angle between the velocity vectors of \(M_{1}\) and \(M_{2}\) after the collision?

Many nuclear collisions are truly elastic. If a proton with kinetic energy \(E_{0}\) collides elastically with another proton at rest and travels at an angle of \(25^{\circ}\) with respect to its initial path, what is its energy after the collision with respect to its original energy? What is the final energy of the proton that was originally at rest?

A method for determining the chemical composition of a material is Rutherford backscattering (RBS), named for the scientist who first discovered that an atom contains a high-density positively charged nucleus, rather than having positive charge distributed uniformly throughout (see Chapter 39 ). In RBS, alpha particles are shot straight at a target material, and the energy of the alpha particles that bounce directly back is measured. An alpha particle has a mass of \(6.65 \cdot 10^{-27} \mathrm{~kg} .\) An alpha particle having an initial kinetic energy of \(2.00 \mathrm{MeV}\) collides elastically with atom X. If the backscattered alpha particle's kinetic energy is \(1.59 \mathrm{MeV}\), what is the mass of atom \(\mathrm{X}\) ? Assume that atom \(X\) is initially at rest. You will need to find the square root of an expression, which will result in two possible an- swers (if \(a=b^{2},\) then \(b=\pm \sqrt{a}\) ). Since you know that atom \(X\) is more massive than the alpha particle, you can choose the correct root accordingly. What element is atom X? (Check a periodic table of elements, where atomic mass is listed as the mass in grams of 1 mol of atoms, which is \(6.02 \cdot 10^{23}\) atoms.)