Page 26 - [David_Morin]_Problems_and_Solutions_in_Introducto(OSN ASTRO_Neat

1.6. MULTIPLE-CHOICE ANSWERS 17

answer this, you will need to invoke the fact that Q is inversely proportional to L. (This
is believable; doubling the length of the pipe will double the effect of friction between the
fluid and the walls, and thereby halve the flow rate.)

1.7. 1-D collision
If a mass M moving with velocity V collides head-on elastically with a mass m that is
initially at rest, it can be shown (see Problem 6.3) that the final velocities are given by

(M − m)V 2MV
V M = and v m = . (1.8)
M + m M + m
Check the M = m, M ≪ m, and M ≫ m limits of these expressions.
1.8. Atwood’s machine
Consider the Atwood’s machine in Fig. 1.6, consisting of three masses and two frictionless
pulleys. It can be shown that the acceleration of m 2 , with upward taken to be positive, is m 1
given by (just accept this)

4m 2 m 3 + m 1 (m 2 − 3m 3 )
a 2 = −g . (1.9) m 2 m 3
4m 2 m 3 + m 1 (m 2 + m 3 )
Figure 1.6
Find a 2 for the following special cases:

(a) m 1 = 2m 2 = 2m 3
(b) m 2 much larger than both m 1 and m 3
(c) m 2 much smaller than both m 1 and m 3
(d) m 1 ≫ m 2 = m 3
(e) m 1 = m 2 = m 3
1.9. Dropped ball
In Section 1.1.4, we looked at limiting cases of the velocity, given in Eq. (1.3), of a beach
ball dropped from rest. Let’s now look at the height of the ball. If the ball is dropped from
rest at height h, and if the drag force from the air takes the form F d = −bv, then it can be
shown that the ball’s height as a function of time equals
mg ( m ( −bt/m ) )
y(t) = h − t − 1 − e . (1.10)
b b
Find an approximate expression for y(t) in the limit where t is very small (or more pre-
cisely, in the limit where bt/m ≪ 1).

1.6 Multiple-choice answers

1.1. e This is the only choice with dimensions of length.

1.2. d This is the only choice with dimensions of time.

2
1.3. c This is the only choice with units of kg m/s .
1.4. b There are 60 · 60 = 3600 seconds in an hour, so one mile per hour equals

mile 1609 meters
1 = = 0.447 m/s. (1.11)
hour 3600 seconds
A common automobile speed of, say, 60 mph is therefore about 27 m/s. The inverse rela-
tion, going from m/s to mph, is 1 m/s = 2.24 mph.