Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

If a radioactive substance has a half-life of one year, does this
mean that it will be completely decayed after two years? Explain.

2

What is the probability of rolling a pair of dice and getting
"snake eyes," i.e., both dice come up with ones?

3

(a) Extract half-lives directly from the graphs shown in figures
k and l on page 23.

(b) Check that the ratio between these two numbers really is about
10, as calculated in the text based on relativity.

4

Use a calculator to check the approximation that

a^{b} ≈ 1 + b ln a ,

if b << 1, using some arbitrary numbers. Then see how good the
approximation is for values of b that are not quite as small compared
to one.

5

Make up an example of a numerical problem involving a rate of
decay where Δt << t_{1/2}, but the exact expression for the rate of decay
on page 54 can still be evaluated on a calculator without getting
something that rounds off to one. Check that you get approximately
the same result using both methods on page 54 to calculate the
number of decays between t and t + Δt. Keep plenty of significant
figures in your results, in order to show the difference between them.

6

Devise a method for testing experimentally the hypothesis that
a gambler's chance of winning at craps is independent of her previous
record of wins and losses.

7

Refer to the probability distribution for people's heights in
figure f on page 48.

(a) Show that the graph is properly normalized.

(b) Estimate the fraction of the population having heights between
140 and 150 cm. p

√

8

(a) A nuclear physicist is studying a nuclear reaction caused in
an accelerator experiment, with a beam of ions from the accelerator
striking a thin metal foil and causing nuclear reactions when a nucleus
from one of the beam ions happens to hit one of the nuclei in
the target. After the experiment has been running for a few hours,
a few billion radioactive atoms have been produced, embedded in
the target. She does not know what nuclei are being produced, but
she suspects they are an isotope of some heavy element such as Pb,
Bi, Fr or U. Following one such experiment, she takes the target foil
out of the accelerator, sticks it in front of a detector, measures the
activity every 5 min, and makes a graph (figure). The isotopes she
thinks may have been produced are:

isotope

half-life (minutes)

^{211}Pb

36.1

^{214}Pb

26.8

^{214}Bi

19.7

^{223}Fr

21.8

^{239}U

23.5

Which one is it?

(b) Having decided that the original experimental conditions produced
one specific isotope, she now tries using beams of ions traveling
at several different speeds, which may cause different reactions.
The following table gives the activity of the target 10, 20 and 30 minutes
after the end of the experiment, for three different ion speeds.

activity (millions of decays/s) after. . .

10 min

20 min

30 min

first ion speed

1.933

0.832

0.382

second ion speed

1.200

0.545

0.248

third ion speed

6.544

1.296

0.248

Since such a large number of decays is being counted, assume that
the data are only inaccurate due to rounding off when writing down
the table. Which are consistent with the production of a single
isotope, and which imply that more than one isotope was being
created?

9

All helium on earth is from the decay of naturally occurring
heavy radioactive elements such as uranium. Each alpha particle
that is emitted ends up claiming two electrons, which makes it a
helium atom. If the original ^{238}U atom is in solid rock (as opposed
to the earth's molten regions), the He atoms are unable to diffuse
out of the rock. This problem involves dating a rock using the
known decay properties of uranium 238. Suppose a geologist finds
a sample of hardened lava, melts it in a furnace, and finds that it
contains 1230 mg of uranium and 2.3 mg of helium. ^{238}U decays by
alpha emission, with a half-life of 4.5 × 10^{9} years. The subsequent
chain of alpha and electron (beta) decays involves much shorter halflives,
and terminates in the stable nucleus ^{206}Pb. (You may want to
review alpha and beta decay.) Almost all natural uranium is ^{238}U,
and the chemical composition of this rock indicates that there were
no decay chains involved other than that of ^{238}U.

(a) How many alphas are emitted in decay chain of a single ^{238}U
atom?
[Hint: Use conservation of mass.]

(b) How many electrons are emitted per decay chain?
[Hint: Use conservation of charge.]

(c) How long has it been since the lava originally hardened?

√

10

Physicists thought for a long time that bismuth-209 was the
heaviest stable isotope. (Very heavy elements decay by alpha emission
because of the strong electrical repulsion of all their protons.)
However, a 2003 paper by Marcillac et al. describes an experiment
in which bismuth-209 lost its claim to fame - it actually undergoes
alpha decay with a half-life of 1.9 × 10^{19} years.

(a) After the alpha particle is emitted, what is the isotope left over?

(b) Compare the half-life to the age of the universe, which is about
14 billion years.

(c) A tablespoon of Pepto-Bismol contains about 4 × 10^{20} bismuth-
209 atoms. Once you've swallowed it, how much time will it take,
on the average, before the first atomic decay? p

11

A blindfolded person fires a gun at a circular target of radius
b, and is allowed to continue firing until a shot actually hits it. Any
part of the target is equally likely to get hit. We measure the random
distance r from the center of the circle to where the bullet went in.

(a) Show that the probability distribution of r must be of the form
D(r) = kr, where k is some constant. (Of course we have D(r) = 0
for r > b.)
(b) Determine k by requiring D to be properly normalized.

(c) Find the average value of r.

(d) Interpreting your result from part c, how does it compare with
b/2? Does this make sense? Explain.

√ d) ∫

12

We are given some atoms of a certain radioactive isotope,
with half-life t_{1/2}. We pick one atom at random, and observe it for
one half-life, starting at time zero. If it decays during that one-halflife
period, we record the time t at which the decay occurred. If it
doesn't, we reset our clock to zero and keep trying until we get an
atom that cooperates. The final result is a time 0 ≤ t ≤ t_{1/2}, with a
distribution that looks like the usual exponential decay curve, but
with its tail chopped off.

(a) Find the distribution D(t), with the proper normalization.

(b) Find the average value of t.

(c) Interpreting your result from part b, how does it compare with
t_{1/2}/2? Does this make sense? Explain.

√ ∫

13

The speed, v, of an atom in an ideal gas has a probability
distribution of the form

D(v) = bve^{-cv2} ,

where 0 ≤ v < ∞, c relates to the temperature, and b is determined
by normalization.

(a) Sketch the distribution.

(b) Find b in terms of c.

(c) Find the average speed in terms of c, eliminating b. (Don't try
to do the indefinite integral, because it can't be done in closed form.
The relevant definite integral can be found in tables or done with
computer software.)

√ ∫

14

Neutrinos interact so weakly with normal matter that, of
the neutrinos from the sun that enter the earth from the day side,
only about 10^{-10} of them fail to reemerge on the night side. From
this fact, estimate the thickness of matter, in units of light-years,
that would be required in order to block half of them. This "halfdistance"
is analogous to a half-life for radioactive decay.