Chapter 8

(III) Early test flights for the space shuttle used a "glider" (mass of 980 kg including pilot). After a horizontal launch at 480 \(\mathrm{km} / \mathrm{h}\) at a height of \(3500 \mathrm{m},\) the glider eventually landed at a speed of 210 \(\mathrm{km} / \mathrm{h}\) . (a) What would its landing speed have been in the absence of air resistance? (b) What was the average force of air resistance exerted on it if it came in at a constant glide angle of \(12^{\circ}\) to the Earth's surface?

(I) For a satellite of mass \(m_{\mathrm{s}}\) in a circular orbit of radius \(r_{\mathrm{S}}\) around the Earth, determine \((a)\) its kinetic energy \(K,\) (b) its potential energy \(U(U=0\) at infinity \(),\) and \((c)\) the ratio \(K / U\)

(1) Jill and her friends have built a small rocket that soon after lift-off reaches a speed of 850 \(\mathrm{m} / \mathrm{s} .\) How high above the Earth can it rise? Ignore air friction.

(I) The escape velocity from planet \(A\) is double that for planet B. The two planets have the same mass. What is the ratio of their radii, \(r_{\mathrm{A}} / r_{\mathrm{B}}\) ?

(1) The escape velocity from planet A is double that for planet B. The two planets have the same mass. What is the ratio of their radii, \(r_{A} / r_{\mathrm{B}} ?\)

(II) Determine the escape velocity from the Sun for an object \((a)\) at the Sun's surface \(\left(r=7.0 \times 10^{5} \mathrm{~km}\right.\), \(\left.M=2.0 \times 10^{30} \mathrm{~kg}\right),\) and \((b)\) at the average distance of the Earth \(\left(1.50 \times 10^{8} \mathrm{~km}\right)\). Compare to the speed of the Earth in its orbit.

(II) Two Earth satellites, \(A\) and \(B\), each of mass \(m=950 \mathrm{~kg}\), are launched into circular orbits around the Earth's center. Satellite A orbits at an altitude of \(4200 \mathrm{~km},\) and satellite \(\mathrm{B}\) orbits at an altitude of \(12,600 \mathrm{~km}\). ( \(a\) ) What are the potential energies of the two satellites? (b) What are the kinetic energies of the two satellites? (c) How much work would it require to change the orbit of satellite A to match that of satellite \(\mathrm{B} ?\)

(II) Show that the escape velocity for any satellite in a circular orbit is \(\sqrt{2}\) times its velocity.

(II) \((a)\) Show that the total mechanical energy of a satellite (mass \(m\) ) orbiting at a distance \(r\) from the center of the Earth (mass \(M_{\mathrm{E}}\) ) is $$ E=-\frac{1}{2} \frac{G m M_{\mathrm{E}}}{r} $$ if \(U=0\) at \(r=\infty\). (b) Show that although friction causes the value of \(E\) to decrease slowly, kinetic energy must actually increase if the orbit remains a circle.

(1I) \((a)\) Show that the total mechanical energy of a satellite mass \(m\) orbiting at a distance \(r\) from the center of the Earth (mass \(M_{E} )\) is $$E=-\frac{1}{2} \frac{G m M_{\mathrm{E}}}{r}$$ if \(U=0\) at \(r=\infty .(b)\) Show that although friction causes the value of \(E\) to decrease slowly, kinetic energy must actu- ally increase if the orbit remains a circle.

(II) Take into account the Earth's rotational speed (1 rev/day) and determine the necessary speed, with respect to Earth, for a rocket to escape if fired from the Earth at the equator in a direction \((a)\) eastward; \((b)\) westward; \((c)\) vertically upward.

(II) ( \(a\) ) Determine a formula for the maximum height \(h\) that a rocket will reach if launched vertically from the Earth's surface with speed \(v_{0}\left(

(II) A meteorite has a speed of \(90.0 \mathrm{~m} / \mathrm{s}\) when \(850 \mathrm{~km}\) above the Earth. It is falling vertically (ignore air resistance) and strikes a bed of sand in which it is brought to rest in \(3.25 \mathrm{~m}\). (a) What is its speed just before striking the sand? (b) How much work does the sand do to stop the meteorite (mass \(=575 \mathrm{~kg}\) )? (c) What is the average force exerted by the sand on the meteorite? (d) How much thermal energy is produced?

(II) How much work would be required to move a satellite of mass \(m\) from a circular orbit of radius \(r_{1}=2 r_{\mathrm{E}}\) about the Earth to another circular orbit of radius \(r_{2}=3 r_{\mathrm{E}} ?\) \(\left(r_{\mathrm{E}}\right.\) is the radius of the Earth.)

(II) A NASA satellite has just observed an asteroid that is on a collision course with the Earth. The asteroid has an estimated mass, based on its size, of \(5 \times 10^{9} \mathrm{~kg}\). It is approaching the Earth on a head-on course with a velocity of \(660 \mathrm{~m} / \mathrm{s}\) relative to the Earth and is now \(5.0 \times 10^{6} \mathrm{~km}\) away. With what speed will it hit the Earth's surface, neglecting friction with the atmosphere?

(II) A sphere of radius \(r_{1}\) has a concentric spherical cavity of radius \(r_{2}\) (Fig. 40\()\) . Assume this spherical shell of thickness \(r_{1}-r_{2}\) is uniform and has a total mass \(M .\) Show that the gravitational potential energy of a mass \(m\) at a distance \(r\) from the center of the shell \(\left(r>r_{1}\right)\) is given by $$U=-\frac{G m M}{r}$$

(III) To escape the solar system, an interstellar spacecraft must overcome the gravitational attraction of both the Earth and Sun. Ignore the effects of other bodies in the solar system. \((a)\) Show that the escape velocity is $$v=\sqrt{v_{\mathrm{E}}^{2}+\left(v_{\mathrm{S}}-v_{0}\right)^{2}}=16.7 \mathrm{km} / \mathrm{s}$$ where: \(v_{\mathrm{E}}\) is the escape velocity from the Earth (Eq. 19); \(v_{\mathrm{S}}=\sqrt{2 G M_{\mathrm{S}} / r_{\mathrm{SE}}}\) is the escape velocity from the gravitational field of the Sun at the orbit of the Earth but far from the Earth's influence \(\left(r_{\mathrm{SE}}\) is the Sun-Earth distance); and \(v_{0}\) is \right. the Earth's orbital velocity about the Sun. (b) Show that the energy required is \(1.40 \times 10^{3} \mathrm{J}\) per kilogram of spacecraft mass [Hint. Write the energy equation for escape from Earth with \(v^{\prime}\) as the velocity, relative to Earth, but far from Earth; then let \(v^{\prime}+v_{0}\) be the escape velocity from the Sun. $$v_{\mathrm{esc}}=\sqrt{2 G M_{\mathrm{E}} / r_{\mathrm{E}}}=1.12 \times 10^{4} \mathrm{m} / \mathrm{s}$$

(I) How long will it take a 1750-W motor to lift a \(335-\mathrm{kg}\) piano to a sixth-story window \(16.0 \mathrm{~m}\) above?

\(\begin{array}{l}{\text { (I) How long will it take a } 1750-\mathrm{W} \text { motor to lift a } 335 \text { -kg }} \\ {\text { piano to a sixth-story window } 16.0 \mathrm{m} \text { above? }}\end{array}\)

(I) If a car generates 18 hp when traveling at a steady \(95 \mathrm{~km} / \mathrm{h},\) what must be the average force exerted on the car due to friction and air resistance?

(I) If a car generates 18 \(\mathrm{hp}\) when traveling at a steady \(95 \mathrm{km} / \mathrm{h},\) what must be the average force exerted on the car due to friction and air resistance?

(I) An \(85-\mathrm{kg}\) football player traveling \(5.0 \mathrm{~m} / \mathrm{s}\) is stopped in \(1.0 \mathrm{~s}\) by a tackler. ( \(a\) ) What is the original kinetic energy of the player? \((b)\) What average power is required to stop him?

(I) An 85 -kg football player traveling 5.0 \(\mathrm{m} / \mathrm{s}\) is stopped in 1.0 \(\mathrm{s}\) by a tackler. (a) What is the original kinetic energy of the player? \((b)\) What average power is required to stop him?

(II) A driver notices that her \(1080-\mathrm{kg}\) car slows down from 95 \(\mathrm{km} / \mathrm{h}\) to 65 \(\mathrm{km} / \mathrm{h}\) in about 7.0 \(\mathrm{s}\) on the level when it is in neutral. Approximately what power (watts and hp) is needed to keep the car traveling at a constant 80 \(\mathrm{km} / \mathrm{h} ?\)

(II) How much work can a 3.0 -hp motor do in \(1.0 \mathrm{~h} ?\)

(II) An outboard motor for a boat is rated at 55 hp. If it can move a particular boat at a steady speed of \(35 \mathrm{~km} / \mathrm{h}\), what is the total force resisting the motion of the boat?

(II) A \(1400-\mathrm{kg}\) sports car accelerates from rest to \(95 \mathrm{~km} / \mathrm{h}\) in \(7.4 \mathrm{~s}\). What is the average power delivered by the engine?

(II) A 1400 -kg sports car accelerates from rest to 95 \(\mathrm{km} / \mathrm{h}\) in 7.4 \(\mathrm{s} .\) What is the average power delivered by the engine?

(II) During a workout, football players ran up the stadium stairs in \(75 \mathrm{~s}\). The stairs are \(78 \mathrm{~m}\) long and inclined at an angle of \(33^{\circ} .\) If a player has a mass of \(92 \mathrm{~kg},\) estimate his average power output on the way up. Ignore friction and air resistance.

(II) A pump lifts \(21.0 \mathrm{~kg}\) of water per minute through a height of \(3.50 \mathrm{~m}\). What minimum output rating (watts) must the pump motor have?

(II) A ski area claims that its lifts can move 47,000 people per hour. If the average lift carries people about \(200 \mathrm{~m}\) (vertically) higher, estimate the maximum total power needed.

(II) A 75 -kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a \(23^{\circ}\) hill. The skier is pulled a distance \(x=220 \mathrm{~m}\) along the incline and it takes 2.0 min to reach the top of the hill. If the coefficient of kinetic friction between the snow and skis is \(\mu_{\mathrm{k}}=0.10,\) what horsepower engine is required if 30 such skiers (max) are on the rope at one time?

(II) \(\mathrm{A} 75\) -kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a \(23^{\circ}\) hill. The skier is pulled a distance \(x=220 \mathrm{m}\) along the incline and it takes 2.0 min to reach the top of the hill. If the coefficient of kinetic friction between the snow and skis is \(\mu_{k}=0.10,\) what horsepower engine is required if 30 such skiers \((\max )\) are on the rope at one time?

(III) The position of a 280 -g object is given (in meters) by \(x=5.0 t^{3}-8.0 t^{2}-44 t,\) where \(t\) is in seconds. Determine the net rate of work done on this object \((a)\) at \(t=2.0 \mathrm{~s}\) and (b) at \(t=4.0 \mathrm{~s}\). (c) What is the average net power input during the interval from \(t=0 \mathrm{~s}\) to \(t=2.0 \mathrm{~s},\) and in the interval from \(t=2.0 \mathrm{~s}\) to \(4.0 \mathrm{~s} ?\)

(III) A bicyclist coasts down a \(6.0^{\circ}\) hill at a steady speed of \(4.0 \mathrm{~m} / \mathrm{s}\). Assuming a total mass of \(75 \mathrm{~kg}\) (bicycle plus rider), what must be the cyclist's power output to climb the same hill at the same speed?

(1II) A bicyclist coasts down a \(6.0^{\circ}\) hill at a steady speed of 4.0 \(\mathrm{m} / \mathrm{s} .\) Assuming a total mass of 75 \(\mathrm{kg}\) (bicycle plus rider), what must be the cyclist's power output to climb the same hill at the same speed?

(II) Draw a potential energy diagram, \(U\) vs. \(x\), and analyze the motion of a mass \(m\) resting on a frictionless horizontal table and connected to a horizontal spring with stiffness constant \(k\). The mass is pulled a distance to the right so that the spring is stretched a distance \(x_{0}\) initially, and then the mass is released from rest.

(1I) Draw a potential energy diagram, \(U\) vs. \(x,\) and analyze the motion of a mass \(m\) resting on a frictionless horizontal table and connected to a horizontal spring with stiffness constant \(k\) . The mass is pulled a distance to the right so that the spring is stretched a distance \(x_{0}\) initially, and then the mass is released from rest.

(III) The potential energy of the two atoms in a diatomic (two-atom) molecule can be written $$ U(r)=-\frac{a}{r^{6}}+\frac{b}{r^{12}} $$ where \(r\) is the distance between the two atoms and \(a\) and \(b\) are positive constants. (a) At what values of \(r\) is \(U(r)\) a minimum? A maximum? (b) At what values of \(r\) is \(U(r)=0 ?\) (c) Plot \(U(r)\) as a function of \(r\) from \(r=0\) to \(r\) at a value large enough for all the features in \((a)\) and \((b)\) to show. ( \(d\) ) Describe the motion of one atom with respect to the second atom when \(E<0,\) and when \(E>0 .(e)\) Let \(F\) be the force one atom exerts on the other. For what values of \(r\) is \(F>0, F<0, F=0 ?\) (f) Determine \(F\) as a function of \(r\).

What is the average power output of an elevator that lifts \(885 \mathrm{~kg}\) a vertical height of \(32.0 \mathrm{~m}\) in \(11.0 \mathrm{~s} ?\)

A projectile is fired at an upward angle of \(48.0^{\circ}\) from the top of a 135 -m-high cliff with a speed of \(165 \mathrm{~m} / \mathrm{s}\). What will be its speed when it strikes the ground below? (Use conservation of energy.)

Water flows over a dam at the rate of \(580 \mathrm{~kg} / \mathrm{s}\) and falls vertically \(88 \mathrm{~m}\) before striking the turbine blades. Calculate (a) the speed of the water just before striking the turbine blades (neglect air resistance), and (b) the rate at which mechanical energy is transferred to the turbine blades, assuming \(55 \%\) efficiency.

A bicyclist of mass \(75 \mathrm{~kg}\) (including the bicycle) can coast down a \(4.0^{\circ}\) hill at a steady speed of \(12 \mathrm{~km} / \mathrm{h}\). Pumping hard, the cyclist can descend the hill at a speed of \(32 \mathrm{~km} / \mathrm{h}\). Using the same power, at what speed can the cyclist climb the same hill? Assume the force of friction is proportional to the square of the speed \(v\), that is, \(F_{\mathrm{fr}}=b v^{2}\), where \(b\) is a constant.

A ball is attached to a horizontal cord of length \(\ell\) whose other end is fixed, Fig. \(8-42 .(a)\) If the ball is released, what will be its speed at the lowest point of its path? \((b)\) A peg is located a distance \(h\) directly below the point of attachment of the cord. If \(h=0.80 \ell,\) what will be the speed of the ball when it reaches the top of its circular path about the peg?

Show that on a roller coaster with a circular vertical loop (Fig, 43), the difference in your apparent weight at the top of the loop and the bottom of the loop is \(6 g^{\prime} s-\) that is six times your weight. Ignore friction. Show also that as long as your speed is above the minimum needed, this answer doesn't depend on the size of the loop or how fast you go through it.

A \(65-\mathrm{kg}\) hiker climbs to the top of a 4200 -m-high mountain. The climb is made in \(5.0 \mathrm{~h}\) starting at an elevation of \(2800 \mathrm{~m} .\) Calculate \((a)\) the work done by the hiker against gravity, (b) the average power output in watts and in horsepower, and (c) assuming the body is \(15 \%\) efficient, what rate of energy input was required.

A 65 -kg hiker climbs to the top of a 4200 -m-high mountain. The climb is made in 5.0 h starting at an elevation of 2800 m. Calculate \((a)\) the work done by the hiker against gravity, \((b)\) the average power output in watts and in horsepower, and \((c)\) assuming the body is 15\(\%\) efficient, what rate of energy input was required.

The nuclear force between two neutrons in a nucleus is described roughly by the Yukawa potential $$ U(r)=-U_{0} \frac{r_{0}}{r} e^{-r / r_{0}} $$ where \(r\) is the distance between the neutrons and \(U_{0}\) and \(r_{0}\left(\approx 10^{-15} \mathrm{~m}\right)\) are constants. (a) Determine the force \(F(r)\) (b) What is the ratio \(F\left(3 r_{0}\right) / F\left(r_{0}\right) ?\) (c) Calculate this same ratio for the force between two electrically charged particles where \(U(r)=-C / r,\) with \(C\) a constant. Why is the Yukawa force referred to as a "short-range" force?

A fire hose for use in urban areas must be able to shoot a stream of water to a maximum height of \(33 \mathrm{~m}\). The water leaves the hose at ground level in a circular stream \(3.0 \mathrm{~cm}\) in diameter. What minimum power is required to create such a stream of water? Every cubic meter of water has a mass of \(1.00 \times 10^{3} \mathrm{~kg}\).

A 16 -kg sled starts up a \(28^{\circ}\) incline with a speed of \(2.4 \mathrm{~m} / \mathrm{s}\). The coefficient of kinetic friction is \(\mu_{k}=0.25 .\) (a) How far up the incline does the sled travel? ( \(b\) ) What condition must you put on the coefficient of static friction if the sled is not to get stuck at the point determined in part \((a) ?\) (c) If the sled slides back down, what is its speed when it returns to its starting point?