fisica1-youn-e-freedman-exercicios-resolvidos

8.80: a) From the derivation in Sec. 8.4 of the text we have
V
A
=
M
M
A
A
− M
+ M
B
B
V
0
and
V
B
2M
A
=
M + M
A
B
The ratio of the kinetic energies of the two particles after the collision is
1
2
1
2
M V
M
2
A A
2
BVB
or KE
A
M
=
M
= KE
A
B
B
⎛V
⎜
⎝ V
A
B
⎞
⎟
⎠
2
( M
A
− M
4M
M
A
M
=
M
B
B
)
2
A
B
⎛ M
A
− M
⎜
⎝ 2M
A
B
⎞
⎟
⎠
2
( M
A
− M
=
4M
M
A
B
B
)
2
i.e.,
b) i) For M = M KE 0;
i.e., the two objects simply exchange kinetic energies.
, A B A
=
ii) For M = 5M
,
KE
KE
B
A
B
2
(4M
B
)
=
4(5M
)( M
A
=
M
A
gets 4/9 or 44% of the total.
c) We want
B
B
)
4
5
KE 2 2
A
( M
A
− M
B
) M
A
− 2M
AM
+ B
KE
B
= 1 =
4M
A
M
B
=
4M
A
M
B
M
2
B
which reduces to
2
2
M
A
− 6M
AM
B
+ M
B
= 0,
from which, using the quadratic formula, we get the two possibilities
M = 5.83
M and
A
B
M = 0.172
M
A
B.