fisica1-youn-e-freedman-exercicios-resolvidos

5.102: (a) The retarding force of the surface is the only horizontal force acting. Thus
1 2
Fnet
FR
− kv dv
a = = = =
m m m dt
dv k
= − dt
1 2
v m
v
∫
v0
dv
=
1 2
v
1 2 v kt
2v
v
= −
0
m
which gives
1 2 2 2
v0
kt k t
v = v0
− +
2
m 4m
For the rock’s position:
1 2 2 2
dx v0
kt k t
= v0
− +
2
dt m 4m
1 2
2 2
v0
ktdt k t dt
dx = v0dt
− +
2
m 4m
and integrating gives
1 2 2 2 3
v0
kt k t
x = v0t
− +
2
2m
12m
(b)
1 2 2 2
v0
kt k t
v = 0 = v0
− +
2
m 2m
This is a quadratic equation in t; from the quadratic formula we can find the single
solution:
1 2
2mv0
t =
k
(c) Substituting the expression for t into the equation for x:
1 2 1 2 2
2 3 3 2
2mv0
v0
k 4m v0
k 8m v0
x = v0
⋅ − ⋅ + ⋅
2
2 3
k 2m
k 12m
k
3 2
2mv0
=
3k
k
m
t
∫
0
dt