Environmental Engineering Reference
In-Depth Information
4.8 It will help if you draw a diagram. The speed of the ball before the collision
with the cushion is
v
i
, after it is
v
f
. The collision causes a change in linear
momentum
p
=
m(v
i
+
v
f
),
and an angular impulse
p (h
R).
If the ball is not to skid or hop, then we
must have rolling without slipping both before and after the collision, so
−
I
R
(v
i
p (h
−
R)
=
+
v
f
).
2
5
mR
2
Now substitute for
p
and
I
=
and solve to obtain
h
=
7
R/
5.
PROBLEMS 5
5.1 Need to figure out the speed relative to the ground. In (a) this is 2 ms
−
1
whilst in (b) it is 0.6 ms
−
1
. Hence the time taken is (a) 20
.
0
/
2
.
0
=
10
.
0s
33
.
3s.
5.2 We can prove this by considering the light to move along the
x
-axis accord-
ing to
x
1
=
and (b) 20
.
0
/
0
.
6
=
ct
whilst the observer in
S
moves according to
x
2
=
X
0
−
ct
where
X
0
is just the position at
t
=
0. The distance between the two is just
x
2
−
x
1
=
X
0
−
vt
−
ct
and the relative speed is the rate of change of this
distance, i.e.
d
(x
2
−
x
1
)
=−
(v
+
c).
d
t
The sign just tells us the distance is decreasing. Notice that no particles
of matter are travelling at this speed (in
S
): it is the relative speed of two
different things.
PROBLEMS 6
6.1
t
=
γt
0
with
t
0
=
30 mins. Since
γ
=
2
.
72 the elapsed time is 81.2
mins.
6.2
γ
=
3
.
20 and hence in the lab frame the half-life is extended to 5
.
76
×
10
−
8
s. Distance travelled is thus 16
.
4m.
6.3
γ
0
.
998
c
.
6.4 If the spacecraft was not sufficiently high we would need to account for the
fact that it moves a significant distance over the time the searchlight is turned
on. At ground level, we need to add on the time taken for light to travel the
extra distance, i.e. total time is
(
0
.
190
=
190
/
12. Solve for
v
=
+
×
=
0
.
380 s. Note
that the key word in the previous question is 'see'. It implies an observation
using your eyes and hence the need to make this correction. Usually we speak
of the intervals between events defined using a network of clocks stationary
in some frame: those intervals do not depend upon where the observer is nor
whether they have eyes or not.
0
.
190
0
.
998
)
s
