215

Solutions to Selected

Problems

Chapter 0

6.

134mg

×

10

–3

g

1mg

×

10

–3

kg

1g

=1.34

×

10

–4

kg

8. (a) Let’s do 10.0 g and 1000 g. The arithmetic

mean is 505 grams. It comes out to be 0.505 kg,

which is consistent. (b) The geometric mean comes

out to be 100 g or 0.1 kg, which is consistent. (c) If

we multiply meters by meters, we get square

meters. Multiplying grams by grams should give

square grams! This sounds strange, but it makes

sense. Taking the square root of square grams (g

2

)

gives grams again. (d) No. The superduper mean of

two quantities with units of grams wouldn’t even be

something with units of grams! Related to this

shortcoming is the fact that the superduper mean

would fail the kind of consistency test carried out in

the first two parts of the problem.

Chapter 1

10.

1mm

2

×

1cm

10mm

2

=10

–2

cm

2

11. The bigger scope has a diameter that’s ten times

greater. Area scales as the square of the linear

dimensions, so its light-gathering power is a hun-

dred times greater (10x10).

12. Since they differ by two steps on the Richter

scale, the energy of the bigger quake is 10000 times

greater. The wave forms a hemisphere, and the

surface area of the hemisphere over which the

energy is spread is proportional to the square of its

radius. If the amount of vibration was the same, then

the surface areas much be in the ratio of 10000:1,

which means that the ratio of the radii is 100:1.

Chapter 2

4. 1 light-year= v

.

t

=

3.0x10

8

m/s

1year

×

365days

1year

24hours

1day

3600s

1hour

=

9.5x10

15

m

5. Velocity is relative, so having to lean tells you

nothing about the train’s velocity. Fullerton is moving

at a huge speed relative to Beijing, but that doesn’t

produce any noticeable effect in either city. The fact

that you have to lean tells you that the train is accel-

erating.

7. To the person riding the moving bike, bug A is

simply going in circles. The only difference between

the motions of the two wheels is that one is traveling

through space, but motion is relative, so this doesn’t

have any effect on the bugs. It’s equally hard for each

of them.

10. In one second, the ship moves v meters to the

east, and the person moves v meters north relative to

the deck. Relative to the water, he traces the diago-

nal of a triangle whose length is given by the

Pythagorean theorem, (v

2

+v

2

)

1/2

=2

1/2

v. Relative to the

water, he is moving at a 45-degree angle between

north and east.

Chapter 3

14.

x

v

t

t

15. Taking g to be 10 m/s, the bullet loses 10 m/s of

speed every second, so it will take 10 s to come to a

stop, and then another 10 s to come back down, for a

total of 20 s.

16.

.

x=

1

2

at

2

, so for a fixed value of

.

x, we have

t

.

1/a

. Decreasing a by a factor of 3 means that t

will increase by a factor of

3

.

Solutions to Selected Problems