978-0073380308 Chapter 5 Solution Manual Part 29

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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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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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

px21dx Dln xCpx21CC:

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

rDRrrP

✓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⌘21D!0dt )Zr

r0

r0⌘21DZt

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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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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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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