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Problem 5.185
A
31;000 lb
truck
A
and a
3970 lb
sports car
B
collide at an inter-
section. At the moment of the collision, the truck and the sports
car are traveling with speeds
v
AD60 mph
and
v
BD50 mph
, re-
spectively. Assume that the entire intersection forms a horizontal
surface. Letting the line of impact be parallel to the ground and
rotated counterclockwise by
˛D20ı
with respect to the preimpact
velocity of the truck, determine the postimpact velocities of
A
and
B
if the contact between
A
and
B
is frictionless and the COR
eD0:1
.
Furthermore, assuming that the truck and the car slide after impact
and that the coefficient of kinetic friction is
kD0:7
, determine
the position at which
A
and
B
come to a stop relative to the position
they occupied at the instant of impact.
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Dynamics 2e 1131
Computation. Substituting Eqs. (6) and (8) into Eqs. (2) and (3), we have
EvC
BD.25:49 O{C67:33 O|/mph D.37:39 O{C98:76 O|/ft=s:
Right after impact,
A
moves along the postimpact direction of the velocity of
A
until the kinetic friction force
due to sliding will cause
A
to stop. The same happens to
B
. Letting
¿
and
¡
denote the positions right after
impact and the final positions of Aand B, applying the work-energy principle we must have
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permission of McGraw-Hill, is prohibited.
Therefore, using the (full precision) values in eqs. (16), and recalling that
kD0:7
and
gD32:2 ft=s2
, we
can evaluate ErAand ErBto obtain
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1133
Problem 5.186
Consider a collar with mass
m
that is free to slide with no friction along
a rotating arm of negligible mass. The system is initially rotating with
a constant angular velocity
!0
while the collar is kept at a distance
r0
from the
´
axis. At some point, the restraint keeping the collar in
place is removed so that the collar is allowed to slide. Determine the
expression for the moment that you need to apply to the arm, as a
function of time, to keep the arm rotating at a constant angular velocity
while the collar travels toward the end of the arm. Hint:
Z1
px21dx Dln xCpx21CC:
Solution
Force Laws.
The force law for this problem must relate the applied moment
M
and the normal force
N
.
This relation has already been found and is given in Eq. (1)
Kinematic Equations. The kinematic equations for this problem are
ErDrOur;EaDarOurCa✓Ou✓;a
rDRrrP
✓2;a
✓DrR
✓C2PrP
✓;P
✓D!0Dconstant:(4)
Equation (6) indicates that to obtain Mas a function of time we need to have both rand Pras a functions of
time. To obtain these expressions we begin by substituting the third and last of Eqs. (4) into Eq. (3), to obtain
where we have used the relation
RrDPrd Pr=dr
. Integrating both sides of Eq. (7) with appropriate limits of
integration we have
0Prd PrDZr
r0
where we have accounted for the fact that, in this problem,
Pr>0
. Next, recalling that
PrDdr=dt
, we can
rewrite the final result in Eq. (8) as
r0
r0⌘21D!0dt )Zr
r0
r0
r0⌘21DZt
0
r0Cs✓r
r0◆2
r0Cs✓r
r0◆2
which can be solved for rto obtain
where
Pr
was obtained by differentiating the expression for
r
with respect to time. Finally, substituting the
above results in Eq. (6) and simplifying, we obtain
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1135
Problem 5.187
A satellite is launched parallel to the Earth’s surface at an altitude of
450 mi
with a speed of
17;500 mph
. Determine the apogee altitude
hAabove the Earth’s surface, as well as the period of the satellite.
Solution
The satellite will orbit the Earth along an elliptical orbit. The launch conditions corresponds to the satellite
being at the perigee of the elliptical orbit. Recalling that the radius of Earth is
reD3959 mi D3959.5280/ ft
,
then the radius at perigee is
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Problem 5.188
A spacecraft is traveling at
19;000 mph
parallel to the surface
of the Earth at an altitude of
250 mi
, when it fires a retrorocket
to transfer to a different orbit. Determine the change in speed
v
necessary for the spacecraft to reach a minimum altitude
of
110 mi
during the ensuing orbit. Assume that the change in
speed is impulsive; that is, it occurs instantaneously.
Solution
The spacecraft intends to transfer onto an orbit with a radius that is smaller than that of its initial orbit.
Therefore, when the spacecraft fires its retrorockets it occupies the apogee of the ensuing elliptical transfer
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Dynamics 2e 1137
Problem 5.189
The optimal way (from an energy standpoint) to transfer from one
circular orbit about a primary body (in this case, the Sun) to another
circular orbit is via the Hohmann transfer, which involves trans-
ferring from one circular orbit to another using an elliptical orbit
that is tangent to both at the periapsis and apoapsis of the ellipse.
This ellipse is uniquely defined because we know the perihelion
radius
re
(the radius of the inner circular orbit) and the aphelion
radius
rj
(the radius of the outer circular orbit), and therefore we
know the semimajor axis
a
via Eq. (5.88) and the eccentricity
e
via
Eq. (5.87) or Eqs. (5.90). Performing a Hohmann transfer requires
two maneuvers, the first to leave the inner (outer) circular orbit and
enter the transfer ellipse and the second to leave the transfer ellipse
and enter the outer (inner) circular orbit. Assume that the orbits of
Earth and Jupiter are circular, use
150⇥106km
for the radius of
Earth’s orbit, use
779⇥106km
for the radius of Jupiter’s orbit, and
note that the mass of the Sun is 333;000 times that of the Earth.
A space probe
S1
is launched from Earth to Jupiter via a
Hohmann transfer orbit. Determine the change in speed
ve
re-
quired at the radius of Earth’s orbit of the elliptical transfer orbit
(perihelion) and the change in speed
vj
required at the radius of
Jupiter’s orbit (aphelion). In addition, compute the time required
for the orbital transfer. Assume that the changes in speed are
impulsive; that is, they occur instantaneously.
Solution
We begin with the determination of the speed corresponding to a circular orbit with radius equal to that of the
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permission of McGraw-Hill, is prohibited.
Therefore denoting the change in speed at perihelion by
veDvperihelion .vcirc/Earth
, using (the full
precision values of) the results in Eqs. (1) and (3), we have
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Dynamics 2e 1139
Problem 5.190
The optimal way (from an energy standpoint) to transfer from one
circular orbit about a primary body (in this case, the Sun) to another
circular orbit is via the Hohmann transfer, which involves trans-
ferring from one circular orbit to another using an elliptical orbit
that is tangent to both at the periapsis and apoapsis of the ellipse.
This ellipse is uniquely defined because we know the perihelion
radius
re
(the radius of the inner circular orbit) and the aphelion
radius
rj
(the radius of the outer circular orbit), and therefore we
know the semimajor axis
a
via Eq. (5.88) and the eccentricity
e
via
Eq. (5.87) or Eqs. (5.90). Performing a Hohmann transfer requires
two maneuvers, the first to leave the inner (outer) circular orbit and
enter the transfer ellipse and the second to leave the transfer ellipse
and enter the outer (inner) circular orbit. Assume that the orbits of
Earth and Jupiter are circular, use
150⇥106km
for the radius of
Earth’s orbit, use
779⇥106km
for the radius of Jupiter’s orbit, and
note that the mass of the Sun is 333;000 times that of the Earth.
A space probe
S2
is at Jupiter and is required to return to the
radius of Earth’s orbit about the Sun so that it can return samples
taken from one of Jupiter’s moons. Assuming that the mass of the
probe is
722 kg
, determine the change in kinetic energy required at
Jupiter
Tj
for the maneuver at aphelion. In addition, determine
the change in kinetic energy required at Earth
Te
for the perihe-
lion maneuver. Finally, what is the change in potential energy
V
of the spacecraft in going from Jupiter to the Earth?
Solution
We start with computing the speed corresponding to a circular orbit with the same radius as Jupiter’s orbit.
Using Eq. (5.82) on p. 388 of the textbook, we have
E
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permission of McGraw-Hill, is prohibited.
