Physics Homework
A 4,000 kg truck going 30 m/s rear-ends a 1,000 kg VW bug going 10 m/s. After the collision, the truck is going 29 m/s. How fast is the VW going?
14 m/s
At full capacity 11,000,000 kg of water flow over the Hoover Dam every second. It falls 220 m to the turbines below. If all of its potential energy is converted into work, what is the power generated?
2.372 x 10^10 W
At what velocity is the relativistic mass of an object equal three times its rest mass?
2√2/3 c m/s
What is the velocity of a satellite whose orbit skims the surface of the earth?
7,909.5 m/s
In class we discussed the case of an observer (on a skateboard) moving with velocity v relative to a water tank in which water waves - generated by a rock - were traveling with speed v (relative to it). We determined that the wave velocity measured by the observer was larger than v for the part of the wave that moved between the rock and the observer, and less than v for the part of the wave that moved on the other side of the rock (from the perspective of the observer). In the first case we determined that the observer obtained a velocity v+v, whereas in the second case found it to be v−v. If the wavelength and frequency of the wave observed in the frame of the tank are denoted by λ(0) and f(0), describe what these compare to the wavelengths and frequencies determined by the observer on the skateboard for both cases (waves between observer and rock and waves on the "other side" of the rock). Note: There is no numerical solution to this problem, rather, you should look for "larger than" or "smaller than" relations and give explanations for your analysis.
As the wave in the tank move left and right, the moving observer would see that v-v shows a higher f(0) and a shorter λ(0), while v+v would look like a lower f(0) and a longer λ(0).
The active part of a guitar string is 60 cm long. What is the wavelength of the fundamental (n=1)? What is the wavelength of the "third harmonic" (n=3)?
a) 120 cm b) 40 cm
The world line of a certain rocket ship is shown in the follow. ing Minkowski diagram. What is the speed of the rocket, (a) as a fraction of the speed of light, and (b) in m/s?
a) 2/3 c b) 2 x 10^8 m/s
We briefly mentioned in class the difference between mass and weight. Your mass is your resistance to acceleration, but your weight is the force of gravity on you, and is (officially) measured in Newtons. So weight (W) is equal to mass (m) times the acceleration of gravity (a(g)): W = ma(g). On earth, a(g) = g = 9.8 m/s^2. (a) Assuming you have a mass of 60 kg, what is your weight on earth? (b) The sun has a mass of 2.0 ×10^30 kg and radius R = 7.0 ×10^8 m. If you could stand on the surface of the sun, what would you weigh? (c) A neutron star is a type of dead star in which the star has collapsed to a superdense state. A typical neutron star has the same mass as the sun, but a radius of only 10 km! What would you weigh on the surface of a neutron star?
a) 588 N b) 16344.5 N c) 8.0 x 10^13 N
Muons are produced high in the atmosphere (say, 10 km up) by the action of cosmic rays. Their velocities are typically close to the speed of light- say 0.999c. We know that the average lifetime of a muon at rest is 2 x 10^-6 s. Would such a muon ever reach ground level according to a) classical laws b) special relativity? (Hint: calculate how far the muon would travel in each case.)
a) 599.4 m, no b) 13,406.34 m, yes
The acceleration of gravity on the surface of the Saturn is 10.4 m/s2 and it has a mass of 5.7 ×10^26 kg. Calculate the radius and average density of Saturn. How does it compare to water?
a) 60,480,289.22 m b) 615.1 kg/m^3 c) Saturn is less dense than water
Suppose that each of the following objects suddenly became a black hole. Calculate the size of the event horizon for: (a) a person who weighs 60 kg; (b) the moon; (c) the black hole in the center of the Milky Way, with mass of 4.2 million suns; (d) the black hole in the center of M87 with mass of 2.4 billion suns. Note that the sun has a mass of 2 ×10^30 kg.
a) 8.89 x 10^-28 m b) 1.25 x 10^10 m c) 1.089 x 10^-4 m d) 7.11 x 10^12 m
A lump of clay is thrown at a brick wall and sticks. a) What happened to its momentum? b) What happened to its kinetic energy?
a) Momentum is still conserved because the wall is connected to the Earth, so it can absorb the momentum with little change. b) Kinetic energy after the collision is dissipated into heat energy and deformation energy as the clay spreads from its original shape and flattens into the wall.
When a rubber ball is thrown at a wall and bounces off, it is typical for it to bounce off at the same speed with which it approached the wall, conserving its kinetic energy. This behavior is common when some very heavy object (like the wall) is collided withsome much lighter object (like the ball). In these collisions, the speed of the lighter object before and after the collision are the same as measured in the frame of the heavy object, though with opposite signs, while the heavy object itself feels little effect at all (see the image below). Now suppose that you observe the heavy object to be moving at 80 km/hr to the right (call that the positive direction) and the lighter object is moving 50 km/hr to the left before their collision. How fast will the lighter object be moving after the collision as measured in your reference frame? To do this problem, first ask what the collision looks like in the frame of the heavier object, and then translate that answer back to your own frame. Note that this is really a relative velocity problem, not a conservation of energy or momentum problem!
v (b/y)= 210 km/hr
A Lincoln Continental is twice as long as a VW Bectle, when they are at rest. As the Continental overtakes the VW, going through a speed trap, a stationary policeman observes that they both have the same length. The VW is going at half the speed of light. How fast is the Lincoln going? (Leave your answer as a multiple of c.)
√13/4 c m/s
Explain in your own words why the principle of Galilean relativity says that the statements "bodies at rest tend to stay at rest" and "bodies in motion tend to stay in motion" are really just the same statement twice.
"Uniform motion" and "rest" are basically the same concept because speed is a relative concept. Mathematically, outside of any perspective, any object at rest or in motion are the same. Relativity of motion makes uniform motion and rest look different, even though both objects are experiencing the same experience.
The speed of sound is 340 m/s. What is the wavelength of concert "A" (440 Hz)?
0.77 m
How many photons does a 100 watt bulb emit in 1 minute? Assume it is pure yellow light, with a frequency of 5 x 10^14 Hz.
1.81 x 10^22 photons/min
Find the kinetic energy in MeV of a neutron with a measured life span of 2065 s, given its rest energy is 939.6 MeV, and rest life span is 900 s.
1216.26 MeV
When you are moving in a circle, the centripetal acceleration inwards creates a "fake" (or fictitious or inertial) force outward which we call the "centrifugal" force. Inertial forces obey the equation F = −ma (note the sign!). If you have mass 60 kg and are going around a 10 m circle once every 3 seconds, what is the centrifugal force on you?
1316 N (assuming that 10 m is the diameter)
The Big Bang is estimated to have released about 10^68 J of energy. How many stars could be created with a tenth of that energy, assuming that the star's mass average is 4 × 10^30 kg?
2.78 x 10^19 stars
In the movie Interstellar, astronauts visit a planet in orbit around the black hole Gargantua. The black hole has a mass of 100 million suns. On the orbiting planet, one hour equals 7 years back on earth. What is the radius of the planet's orbit?
2.96 x 10^11 m
On your twenty-first birthday you depart for a nearby star, on the express rocket, which travels at 3/S the speed of light. At your destination you immediately catch the return flight (which also goes at 3/5 c), arriving home just in time to celebrate your twenty-fifth birthday. How old is your twin brother (who stayed at home)?
26 years old
By letting the vertical axis be ct, instead of just t, we are, in effect, measuring time in meters. The unit of ct is the distance light travels in one second (a "light-second). What is a light-second, in meters?
3 x 10^8 m
Make a measurement of the speed of electromagnetic waves. (Microwave project)
3.04 x 10^8 m/s |err|= 1.34%
Helium-neon lasers have a wavelength of 6.338 x 10^-7 m. What is the energy of a single photon in the beam?
3.14 x 10^-19 J
An atomic clock placed in an airplane measures a time interval of 3600 s when moves with speed 400 m/s. How much larger a time interval does an identical clock on the ground measure?
3.2 x 10^-9 s
Particle A, of mass m, is at rest when it decays into two identical particles B, each of (rest) mass (2/5)m: A -> B + B. The Bs fly off in opposite directions. What is the speed of each B? [Leave your answer as a multiple of c.]
3/5 c m/s
A rock is thrown straight up with an initial velocity of 30 m/s. How high will it rise before coming back down to Earth?
45.92 m
We will soon be learning about gravity in more detail, but you should be able to answer this already: The earth exerts a force on a 50 kg person of about 500 N. What is the magnitude of the force that the person exerts on the earth? You can look up any necessary data you need on the internet or in your text, but you do not need to know anything about gravity to do this problem or the other gravity problems below!
500 N
Suppose your twin sister goes off to live on the surface of a neutron star for 10 years (as measured by your clocks on earth). When she come back, how much will she have aged? You may assume that a neutron star has the same mass as our sun but has a radius of only 10 km. You may also ignore the effects of the Earth's gravity on your own clock, since they are so small.
8.39 years
An object is dropped from rest from a height of 4 m above the surface of the earth. Ignoring air resistance, what is the speed of the object when it hits the ground?
8.85 m/s
It takes Johnny 7 mins to eat a hot dog, according to his own watch. If Johnny is on a high-speed train, going 90% the speed of light, how long does it take him to each the hotdog according to the observers, who watch him through the dining car window?
963 seconds or 16 minutes
Suppose for a second that gravity itself is really a fictitious force. What would that imply? (In other words, what would need to be true about your motion for that to be so?)
A fictitious force is defined as F (fictitious)= -ma. if gravity was a fictitious force, then objects and people on Earth would be actively accelerating away from the Earth and towards space. Gravity would move oppositely of how we experience it and how it is defined now (towards Earth's center).
After filling your car's tank with gas, you speed down the highway, drive to the top of a hill and take your foot off the gas. You roll down the hill and at the bottom you pressthe brakes to stop the car. List all the transformations that occurred for the energy that was once in the gasoline? Where is all that energy at the end of the trip? (Assume thecar is a typical gas-powered car, not hybrid or electric.)
Chemical energy to gravitational/potential energy to kinetic energy to waste energy (heat/sound)
In class, we learned that the total gravitational energy of a planet in orbit about the Sun is: E = K + U = 1/2 mv^2 − GmM(s)/r where m is the mass of the planet, Ms is the Sun's mass, r is the orbital radius, and v is the speed of the planet in its orbit. Using the results we found in class for v, show that the total energy, E, of the earth in its orbit is negative. (You're not trying to number - find an expression for E that is obviously negative.)
E= -1/2 GmM(S)/r
If the sun suddenly collapsed into a neutron star, what would happen to the Earth's orbit? (Hint: how would the strength of the sun's gravity on the Earth change?)
Earth's orbit would not change immediately after because the sun's mass would stay the same after collapsing. There is no charge in the sun's gravitational pull on the Earth.
Show that E^2(r)-P^2(r)C^2=M^2C^4
Replace E with relative mc^2 and P with relative mv, then solve
This problem will require some research from your part. Your response therefore must include references but do write it "in your own words". So-called "shortwave radios" can be used to transmit over very large distances, even transcontinental. How do they manage to do so? Why don't we use them for all our radio broadcasts?
Shortwave radio waves emit at frequencies of 3-30 MHz. the waves reflect off of the ionosphere and reflect back down to Earth from the sky. Since they are reflected instead of emitted in a straight line on the Earth's surface, shortwave radio can broadcast over longer distances. However, they cannot be used for everything because of disrupting factors such as weather/daylight patterns, distances where the spherical shape of Earth would not reflect correctly, and the strength of the emission.
You are sitting on a stopped train, holding a helium birthday balloon by a string so that it floats above you. The train begins to accelerate forward. What does the balloon do? (To do this problem, use the equivalence principle.)
The balloon goes forward with the train's acceleration as we go backwards. This is because the balloon already disobeys gravity by floating up, so it would go forward as well.
Explain in your own words why it took so long historically for the Law of Inertia to be discovered. Why isn't it obvious?
The law of inertia depends on experiences to prove its existence that cannot be replicated on Earth because of friction. Since the physical laws of nature cannot be tested inside of Earth's laws, it took a long time to realize that any object will indefinitely move or stay still unless acted on by another force. Also, inertial frames of reference are basically just frames where the law of inertia holds true, but without the concept of forces/friction it was difficult to understand dhow an object would love or stay still indefinitely.
On the first day of class, we played a bit with Newton's cradle, a series of identical pendulums that are lined up in a tight row. We saw that if we take a single bob onone end and let it collide with the row of bobs, then a single bob will move away onthe opposite end. Here's a claim: the only way for Newton's cradle to conserve both momentum and energy in the collision of the one bob with the others is for one, and only one, bob to be pushed out the other side. This is not at all obvious. But let's try to convince ourselves this is correct. Imagine that the first bob has mass m and hits the row of bobs with speed v. Clearly, having the last bob fly off with speed v will conserve both momentum and energy (note:the only form of energy relevant to this problem is kinetic energy). But suppose that you wanted two bobs to fly off after the collision. Assume those two bobs have speeds u1 and u2 respectively. Using conservation of momentum and conservation of energy, find two equations for u1 and u2 in terms of v, and solve for each. What do you find?
The scenario in which one bob could make two bobs fly off is mathematically impossible. u1=v and u2=0.
Here is one way to check whether the earth is really a sphere or it is flat, without leaving it. Imagine performing the following task on the surface of the earth: Starting at the equator,you go due north for 1000 miles. Then you turn right and go due east for 1000 miles. Then you turn right and go due south 1000 miles. Then you turn right and go due west 1000 miles. Where do you end up? (No numbers needed, just give your position relative to your startingpoint.) How does that tell you the earth is a sphere? (It might be easiest if you draw a picture for yourself.)
You would end slightly east of your starting point because non-euclidean geometry permits a 270 degree triangle on a sphere. As you travel North, the radius decreases, causing you to travel eastward farther relative to the increasing radius as you go South.
A hydrogen atom undergoes a transition from the state n lite stale n = 1. (a) What was the initial energy of the atom? (b) What is the final energy of the atom? (c) What is the energy of the emitted photon? d) What is the frequency of the emitted photon? (e) Is this photon in the visible region? If not, what sort of radiation is it?
a) -0.852 eV b) -13.61 eV c) 12.77 eV d) 3.09 x 10^15 Hz e) Not visible, ultraviolet
A supernova explosion of a 2 × 10^31 kg star produces 1 × 10^44 J of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio ∆m/m of mass destroyed to the original mass of the star?
a) -1.11 x 10^27 kg, loss b) 5.56 x 10^-5
Uranium-238 spontaneously disintegrates into thorium-234 plus helium-4:18 238U-234Th+4He. The rest masses of these atoms are: 238U: 238.05079 u; 234Th: 234.043 63u; 4He: 4.002 60 u. (The atomic mass unit u is 1.66 x10^-27 kg.) (a) How much rest mass is lost in this decay? (First give your answer in u; then convert to kg.) (b) How much kinetic energy is created in each disintegration? (That's the same as the rest energy lost.) (c) How many disintegrations would it take to generate 1 kwh of electricity? (d) How many kwh could you get altogether if you started with 1 gram of 238U? (Actually, the half-life of 238U is 4.5 x 10^9 years, which is kind of a long time to wait - but there are ways to speed up the process.)
a) 0.00456 u or 7.5696 x 10^-30 kg b) 6.81 x 10^-13 J c) 5.29 x 10^18 disintegrations d) 478,703 kwh
We are used to thinking that when we sit at one location on the earth's surface, we areat rest. But with respect to other frames we are moving. For example, our spot of earth is rotating around the center of the earth once every 24 hours. (a) If the earth has a radius of 3800 miles, what is our speed with respect to the center when we are just sitting? Express your answer in miles/second. (For those of you who want to be picky, you can assume we are sitting on the equator, since people sitting closer to the poles are actually moving more slowly.)(Recall: A circle has circumference 2πr.) (b) The earth is traveling around the sun once peryear, in a circle of radius 90 million miles. What is our "sitting" speed with respect to thesun? (c) Our entire solar system is rotating about the center of our galaxy, the Milky Way, once every 230 million years with a radius of around 1.5 ×1017 miles. What is our "sitting" speed with respect to the galactic center?
a) 0.276 miles/second b) 17.9 mile/second c) 129.8 miles/second
Compute (without using your calculator) the values for γ and γ −1 for the following values of v: v = (0, 10−7c, 10−4c, 0.1c, √5c/3, √3c/2, c). Hint: You can use the binomial expansion for some of these values.
a) 1 b) 1-(5 x 10^-13) c) 1-(5 x 10^-7) d) 1-(5 x 10^-1) e) 1- 5/18 f) 1- 3/8 g) 0
(1 + ε)n ≃1 + nε if |ε|≪1 and |nε|≪1. Use this expansion to find approximate solutions to the following problems without using your calculator, and then plug them into your calculator to find the exact solutions and compare. (a) (1.02)^2 (Hint: you can rewrite this as (1 + 0.02)^2.) (b) (1.02)^3 (c) (1.02)^50 (Do you expect the approximation to work in this case?) (d) √1.02 (e) 4√1.02 (f) 1/1.02 (g) 1/(1.02)^2 (h) 1/√0.96 (i) (1 + 10−10)^2 (Note: don't give your approximate answer as 1 be more precise.) (j) 2/√1.1 Here are a couple optional ones, just to make you think: (k) √4.8 (ℓ) 3√8.24
a) 1.04 b) 1.06 c) 2 (not a good estimate) d) 1.01 e) 1.005 f) 0.98 g) 0.96 h) 1.02 i) 1 j) 1.9 k) 2.2
A typical blue photon has a (relativistic) energy 4.30 x/I0^-19 J. (a) What is is (relativistic) momentum? (b) What is its (rest) mass? (c) What is its speed? (d) Using P(r) = m(r)v, what is its relativistic mass?
a) 1.43 x 10^-27 kgm/s b) 0 kg c) 3 x 10^8 m/s d) 4.77 x 10^-36 kg
The star nearest Earth, Alpha Centauri, is 4.3 light years away. If a spaceship makes the (one-way) trip at a constant velocity of 0.95c, how long does it take according to Earth's clock? How long does it take according ti the clock on the spaceship?
a) 1.43 x 10^8 seconds b) 4.46 x 10^7 seconds
What is the de Broglie wavelength of (a) a baseball with mass 0.15 kg moving at 30 m/s (b) an electron with kinetic energy of 10^−18 J? (The mass of an electron is 9.1×10^−31 kg.)
a) 1.472 x 10^-34 m b) 4.9 x 10^-10 m
Our solar system lives inside the Milky Way galaxy, and orbits the center of thegalaxy at a velocity of 220 km/s. The distance to the center of the galaxy is about 26,000 light-years (1 light-year is 9.5 ×10^15 m). The net gravitational force that we experience from the galaxy is due to the masses of all the stars that lie at radii smaller than our Sun's, and can be treated as if they are all sitting at the galaxy's center; we call this mass "enclosed" by our orbit. (The gravity from the stars outside our orbit all exactly cancels out, and so doesn't affect us.) (a) Given the numbers above, what is the total mass enclosed by our orbit through our galaxy? (b) Assuming the sun to be a pretty average star, with Ms = 2.0 ×10^30 kg, how many stars would you expect to be "enclosed" by our orbit?
a) 1.79 x 10^41 kg b) 8.96 x 10^10 stars
Because the radius of a blackhole grows linearly with the mass, the volume of the a blackhole (which goes like R3) grows much faster than the mass. Thus the density of a black hole actually goes down(!) with mass. Calculate the density (in kg/m3) of a black hole with the mass of (a) our sun, (b) the Milky Way blackhole, (c) the M87 blackhole, and (d) our universe (about 1 mole of suns). How do these compare to the density of water and/or air?
a) 1.83 x 10^19 kgm^3, much denser than air and water b) 1,030,808.637 kgm^3, much denser than air and water c) 3.19 kgm^3, much less dense than water and slightly less dense than air d) 5.03 x 10^-29 kgm^3, much less dense than air and water
Suppose you shake a rope up and down twice a second. What is the period of the resulting wave? What is its frequency?
a) 1/2 seconds b) 2 Hz
(a) A bullet of mass 0.0090 kg is fired from the end of the barrel of a rifle. It exists the riflegoing 1200 m/s. What is the final momentum of the bullet? (b) If the gun exerted a constantforce of 130 N on the bullet while it was in the barrel, how long did it spend in the barrel? (c) If the person firing the gun is standing on frictionless ice and has a mass of 80 kg, how fast will he/she be traveling backwards after the gun is fired?
a) 10.8 kg per m/s b) 0.08 seconds c) 0.135 m/s
An automobile starts from rest and after 3 seconds is moving with a speed of 21 m/s. If acceleration was constant, how far did the automobile move in the first 2 seconds? How far did it move after the third second?
a) 14 m after 2 seconds b) 17.5 m during the final second
You are pulling a child on a sled across a frozen pond (the ice makes it frictionless). The mass of the child-plus-sled is 20 kg, and you exert a force of 40 N for 3 seconds, starting from rest. a) What is the acceleration of the sled? b) How far does the sled go? c) How fast is it going, at the end? d) How much work did you do? e) What is the final kinetic energy of the sled-plus-child? f) What is the power output?
a) 2 m/s^2 b) 9 m c) 6 m/s d) 360 J e) 360 J f) 120 W
Suppose that a person can withstand a tidal force of 2000 N between their head and feet before perishing. Would you perish inside or outside the event horizon for a black hole with mass of the sun? Would you perish inside or outside the event horizon of a black hole whose mass is 100 billion times that of the sun? Assume your head and feet to each have a mass of 10 kg, and your height to be 2 meters.
a) 2.048 x 10^11 N, Inside b) 2.05 x 10^-11 N, outside
What is the kinetic energy of the electron in the ground state of hydrogen? What is its velocity? What percent of the speed of light is this? Would you agree that it is reasonable to use the classical formulas for energy and momentum?
a) 2.18 x 10^-18 J b) 2.19 x 10^6 m/s c) 0.73% d) Yes because the velocity is such a small fraction of c that it would still be reasonable in the range of classical mechanics.
(a) A pendulum of length 0.25 m is held straight out to the side. On the end of the pendulum is a mass m = 0.12 kg. The pendulum is released. How fast is it going at the bottom of its arc? (b) The same pendulum is released from the same position, but this time it is given a push so that it starts with a speed of 1.6 m/s. What is its speed at the bottom of the arc?
a) 2.21 m/s b) 2.73 m/s
An electron in a hydrogen atom makes a transition from the Wilen = 4 to the state n = 2. (a) What is the energy of the emitted photon? b) What is its frequency? (c) What is its wavelength?
a) 2.5505 eV b) 6.17 x 10^14 Hz c) 4.86 x 10^-7 m
By Newton's time, the radius of the earth was fairly well known, RE ≃ 6 ×10^6 m, as was the acceleration of gravity, a(g) ≃ 10 m/s. But of course the earth's mass wasn't known. Nonetheless, people made estimates of the mass based on what they believed the earth was made out of (dirt and rocks). Make your own estimate of the earth's mass (explain how you got it) and use it to derive a value for G(N).
a) 2.71 x 10^24 kg b) G(N)= 1.33 x 10^-10 m^3/kg/s^2
Satellites in geosynchronous orbit fly directly over the equator and stay over the same spot on the surface of the earth at all times. What is their period? What is the radius of their orbit?
a) 24 hours or 86,400 seconds b) 42,240,065.32 m
A car driving north at constant velocity on SE 39th passes Tolman (600 meters south of Woodstock) at 12:07 p.m. and reaches Holgate (2400 meters north of Woodstock) at 12:12 p.m. Find: (a) delta x (in meters), (b) delta t (in seconds), and (c) the velocity of the car (in meters per second).
a) 3,000 m b) 300 seconds c) 10 m/s
An object moving initially with a velocity of 12 m/s is uniformly accelerated at a rate of 3 m/s^2. What is its velocity after 8 seconds of acceleration? If it started at position 8m, what is its position after 8 s?
a) 36 m/s b) 200 m
Helium-neon lasers have a wavelength of 6.328 x 10^-7 m. What is the frequency of this light? What color is it?
a) 4.74 x 10^14 Hz b) orange
Suppose you drop a rock down a well, and 3 seconds later you hear the splash. a) How deep is the well (That is: how far down is the surface of the water? Neglect air resistance and the time it took for the sound to reach you.) b) How fast was the brick going when it hit the water? c) Sound travels at 340 m/s. If we do take into account the time it took the sound to reach you, what is the corrected depth of the well?
a) 44.1 m b) 29.4 m/s c) 40.65 m
a) How much energy must be given to a 1 kg mass to accelerate it from rest to 30 m/s? (b) Once it is going 30 m/s, how much additional energy must be given to accelerate it to 60 m/s?
a) 450 J b) 1350 J
AM radio station KPOJ broadcasts at a frequency of 620 kHz (6.2 x 10^5 Hz). What is the wavelength of the signal? What is the period of the oscillations?
a) 483.5 m b) 1.6 x 10^-6 seconds
Two loudspeakers, mounted 3 m apart on a wall, are driven in unison by the same amplifier, delivering a sustained note with a wavelength of 2 m. You are standing 4 m in front of one of the speakers, as shown in the figure below. a) How far are you from the other speaker? b) How many wavelengths are you from each speaker? c) What do you hear? d) If you move 1.5 m to the right (so you are the same distance from both speakers), what will you hear?
a) 5 m b) 2 λ and 2.5 λ c) Complete destructive interference, hear nothing d) Complete constructive interference, double the volume
Put in the known values of h, k, m, and e to determine (a) the Bohr radius and (b) the ground state energy of hydrogen - first in joules, then convert to electron volts.
a) 5.22 x 10^-11 m b) -13.62 eV
A violin has been tuned so that the velocity of waves on the E string (33 cm long) is 435 m/s. a) What is the wavelength of the fundamental? What is its frequency? b) The vibrating string sets up sound waves in the air. Their frequency is the same as the frequency of the waves on a string (of course), but their wavelength is completely different because the speed of sound in air (340 m/s) is not the same as the speed of waves on the string. Find the wavelength of the resulting sound wave.
a) 66 cm and 659.1 Hz b) 0.52 m
(a) When you jump up in the air, you are pulled back down with constant accelerationg = 9.8 m/s2. While you are up in the air, however, the earth is also being pulled towards youby your gravity. What acceleration does the earth experience towards you while you are in theair? (Assume your mass is 60.0 kg. The mass of the earth is 6.0 ×1024 kg.) (b) Suppose that your speed while you are in the middle of your fall is 12 m/s. What is the earth's speed at thesame moment?
a) 9.8 x 10^-23 m/s b) 1.2 x 10^-22 m/s
In class we derived an expression for the relation of the time interval between the firing of a laser beam to a mirror and its return to the gun as observed from two different frames (A and B) of reference in relative motion with velocity v. The firing of the light beam happened at rest in frame B. The expression we found is ∆TA = γ ∆TB, where ∆TA,B denote the time intervals in frames A and B respectively, and γ is given by γ = (1−(v^2/c^2))^-1/2. a) Make a plot of γ vs v (such as the one we did in class) and thus show that γ ≥ 1. On the same plot, add a plot of γ−1 vs v. b) Discuss what happens to γ when: i) v c, ii) v ∼ c, and iii) v ≥ c. c) Discuss what happens to γ−1 when: i) v c, ii) v ∼ c, and iii) v ≥ c.
a) Exponential growth and decay with an asymptote at c b) i) γ goes to one, ii) γ goes to infinity, iii) γ is not a real number c) i) γ goes to one, ii) γ goes to negative infinity iii) γ is not a real number
There are other geometries that aren't spheres or hyperboloids. (a) Suppose you are standing on a mysterious planet. You take a trip straight due north and end up back where you started after a couples days of walking; obviously the same happens when you go due south. But when you head east or west, you never return to where you began. What might the geometry/shape of this "planet" be? (b) Now suppose to land on another planet with the same north/south behavior as above, but now when you walk east or west, you do return to the same point where you started, but only after several weeks. What is a possible shape forthis planet?
a) Infinite horizontal cylinder b) ellipsoid
The notion of de Broglie wavelength is not invariant under relativity, since an object's momentum depends on its frame. But the maximum velocity that any object can have in any frame is v = c, so a minimum wavelength is given by the expression:λ = h/mc. This is also known as an object's Compton wavelength. We can use the notion of Compton wavelength to define the interface between general relativity and quantum mechanics. In particular, if an object's Compton wavelength is larger than its Schwarzschild radius, it cannot be a black hole, since there is a large probability that it will be sitting outside its own horizon. (a) Is an electron a black hole? (b) What is the minimum mass that a black hole can have?
a) Not a black hole, 2.43 x 10^-12 m >> 1.33 x 10^-57 m b) 3.86 x 10^-8 kg
We mentioned in class that John Michell (and later Pierre-Simon Laplace) proposed that black holes might exist (he called them "dark stars"). Black holes are objects from which even light cannot escape the surface: the escape velocity is faster than the speed of light. Suppose that light leaves the surface of a black hole with velocity v = c, where c is the speed of light, 3.0 ×10^8 m/s. Using the result of problem #8, show that the escape velocity from any object exceeds the speed of light when: R = 2GM/c^2where R is the radius of the object and M is its mass. Then calculate the radius of a black hole whose mass is equal to the mass of the Sun, M = 2.0 ×10^30 kg.
a) Since light cannot escape a black hole, escape velocity> speed of light b) 2, 966.2 m
Apply Mach's principle to the following scenarios and explain how it differs from Newton's view, if it does: (1) in our Universe, two observers are at rest with respect to one another and with respect to the majority of stars in the universe, but then one accelerates away. Which observer will feel an intertial force? (2) Consider the same scenario in a universe that only contains the two observers and nothing else?
a) The accelerating observer would feel an inertial force b) Both obeservers would feel an inertial force c) Newton's view is only relative to viewer, while Mach considers all matter in the universe
A particle starts at x = 2 m, at t = 0s, and moves in the x direction at a constant speed of 1/5 c for 5 x 10^-8 s. It then stops for 2 x 10^-8 s, turns around, and goes back at constant speed to x = 0 m. The entire trip took 12 x 10^-8 s. (a) Sketch the world line for this trip. (b) On your figure, indicate the particle's "future," at time t = 6 x 10^-8 s. (c) For each of the following points in spacetime, say whether it is in the future, the past, or elsewhere, for the particle at t = 6x 10^-8s: (i) x = 6 m, 1 = 2 x 10^-8 s; (i) x = 3 m, t = 10 x 10^-8 s; (iii) x = 2 m, t = 7x 10^-8 s. Plot these points on your graph.
a) Time cone b) On figure c) i) past, ii) future, iii) future
A police car is sitting in the median strip of a highway clocking the speed of cars goingboth east and west on the highway. He clocks car A going 72 mph eastward, and car B going 65 mph westward. (a) What is the velocity of car A in the frame of car B? (We will define eastward to be the positive direction for this problem so we all have the same signs.) (b) What is the velocity of car B in the frame of car A? (c) What is the velocity of the police car in the frame of car A? Please be careful with signs throughout this problem!
a) v (a/b)= 137 mph b) v (b/a)= -137 mph c) v (c/a)= -72 mph
In fact, following the previous problem, the total energy, E, of any object caught in an orbit is always negative - if an object has E > 0, that means is it not "bound" into an orbit, but is "free" to leave the vicinity of the other, more massive object. So consider an object moving through our solar system. At some moment we measure its distance to the Sun to be r and its velocity to be v. What is the lower bound on v such that the object will escape the gravitational pull of the Sun and leave our solar system? (You're deriving a formula here. We call this velocity the "escape velocity".)
v=√(2GM(S)/r)
What is the velocity of a particle that has a kinetic energy equal to its rest energy?
√3/2 c m/s
