160 Solutions Manual
Problem 2.110
The jaguar
A
leaps from
O
at speed
v0
and angle
ˇ
relative to
the incline to intercept the panther
B
at
C
. The distance along
the incline from
O
to
C
is
R
, and the angle of the incline with
respect to the horizontal is ✓.
Derive
v0
as a function of
ˇ
to leap a given distance
R
along with the optimal value of launch angle
ˇ
, i.e., the value
of
ˇ
necessary to leap a given distance
R
with the minimum
v0
. Then plot
v0
as a function of
ˇ
for
gD9:81 m=s2
,
RD7
m, and
✓D25ı
, and find a numerical value of the
optimal
ˇ
and the corresponding value of
v0
for the given set
of parameters.
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of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
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162 Solutions Manual
Problem 2.111
A stomp rocket is a toy consisting of a hose connected to a blast pad (i.e., an air bladder) at one end and to
a short pipe mounted on a tripod at the other end. A rocket with a hollow body is mounted onto the pipe
and is propelled into the air by stomping on the blast pad.
If the rocket can be imparted an initial speed
v0D120 ft=s
, and if the rocket’s landing spot at
B
is
at the same elevation as the launch point, i.e.,
hD0ft
, neglect air resistance and determine the rocket’s
launch angle
✓
such that the rocket achieves the maximum possible range. In addition, compute
R
, the
rocket’s maximum range, and t
f, the corresponding flight time.
Solution
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 163
Problem 2.112
A stomp rocket is a toy consisting of a hose connected to a blast pad (i.e., an air bladder) at one end and to
a short pipe mounted on a tripod at the other end. A rocket with a hollow body is mounted onto the pipe
and is propelled into the air by stomping on the blast pad.
Assuming the rocket can be given an initial speed
v0D120 ft=s
, the rocket’s landing spot at
B
is
10 ft
higher than the launch point, i.e.,
hD10 ft
, and neglecting air resistance, find the rocket’s launch angle
✓
such that the rocket achieves the maximum possible range. In addition, as part of the solution, compute the
corresponding maximum range and flight time. To do this:
(a) Determine the range Ras a function of time.
(b)
Take the expression for
R
found in (a), square it, and then differentiate it with respect to time to find
the flight time that corresponds to the maximum range, and then find that maximum range.
(c) Use the time found in (b) to then find the angle required to achieve the maximum range.
Solution
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permission of McGraw-Hill, is prohibited.
164 Solutions Manual
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 165
Problem 2.113
A trebuchet releases a rock with mass
mD50 kg
at the point
O
. The
initial velocity of the projectile is
Ev0D.45 O{C30 O|/m=s
. If one
were to model the effects of air resistance via a drag force directly
proportional to the projectile’s velocity, the resulting accelerations
in the
x
and
y
directions would be
RxD.⌘=m/ Px
and
RyDg
.⌘=m/ Py
, respectively, where
g
is the acceleration of gravity and
⌘D0:64 kg=s
is a viscous drag coefficient. Find an expression for
the trajectory of the projectile.
Solution
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permission of McGraw-Hill, is prohibited.
166 Solutions Manual
Problem 2.114
Continue Prob. 2.113 and, for the case where
⌘D0:64 kg=s
, deter-
mine the maximum height from the ground reached by the projectile
and the time it takes to achieve it. Compare the result with what you
would obtain in the absence of air resistance.
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permission of McGraw-Hill, is prohibited.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
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168 Solutions Manual
Problem 2.115
Continue Prob. 2.113 and, for the case where
⌘D0:64 kg=s
, deter-
mine
tI
and
xI
, the value of
t
, and the
x
position corresponding to
the projectile’s impact with the ground.
Solution
We begin by working part of the solution to Prob. 2.113. Specifically, we can integrate the
x
and the
y
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Dynamics 2e 169
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permission of McGraw-Hill, is prohibited.