4: Momentum bandits: the slingshot effect

space

We can also treat gravity assist as a perfectly elastic collision (see in2 Physics

@ Preliminary p 66). The planet pulls the probe rather than pushing and the

‘collision’ is gradual, but the conservation of momentum still applies. It is elastic

because gravity conserves mechanical energy—there is no ‘friction’. The planet

is like an enormously heavy cricket bat and the probe is a small, highly elastic

‘superball’. In the bat’s frame of reference, the bat is a stationary ‘immovable’

object (see in2 Physics @ Preliminary p 66), so in a highly elastic collision the

ball’s speed before and after the collision is practically unchanged. In the frame

of reference of the batsman, however, the ball has been given an increase in speed

by the moving bat.

In 1973, Mariner 10 first used gravity assist (from Venus) to achieve

a flyby of Mercury to gather images and other measurements. The Galileo probe

was launched from a Space Shuttle in 1989, reached Jupiter in 1995 and

studied Jupiter’s moons until 2003. To reduce the explosion hazard to the

Shuttle’s astronauts during launch, Galileo’s fuel requirement was decreased

by using gravity assist, once from Venus and twice from Earth, to slingshot

Galileo to Jupiter. The reduction in Earth’s orbital velocity was of the order

of only 10–18 m s–1.



spacecraft’s

velocity outbound

vout = vin



spacecraft’s

trajectory



Jupiter



spacecraft’s

velocity

inbound



resultant Vout



Vout > Vin



Worked example



Jupiter



Question

For gravity assist, the maximum possible speed increase occurs for the (unrealistic) extreme

limit at which the spacecraft executes a nearly 180° turn, parallel to the planet’s orbital

motion (Figure 2.4.2).



Jupiter’s velocity

relative to the Sun

resultant Vin



Figure 2.4.1 Gravitational slingshot in



Show that for this special case, the change in speed of the craft in the Sun’s frame of

reference is twice the orbital speed of the planet Vp. You can assume that in the Sun’s

frame, the probe’s speed is always larger thanVp.



(a) Jupiter’s frame of

reference and (b) the Sun’s

frame of reference



Solution

Be careful! Change in speed Vout – Vin is the change in the magnitude of the velocity, but

it is not the same as the magnitude of change in velocity |Vout – Vin|.

All labelled velocities in Figure 2.4.2 lie in one dimension. Upper case variables denote

quantities in the Sun’s reference frame, and lower case variables the planet’s frame.

As usual, bold = vectors, italic = magnitudes and sign = direction.

Vin, Vout and Vp are respectively the spacecraft’s incoming and outgoing speeds and

the planet’s orbital speed in the Sun’s frame of reference. Because of orbital symmetry, in

the planet’s frame the spacecraft’s incoming and outgoing speeds vin and vout are equal.

Let them both equal v.



a



vout



vin

b



Vout



Using the Galilean transformation formula vB (rel. to A) = vB ‑ vA to convert velocities

in Figure 2.4.2a from the planet’s frame and using the sign convention + → :





vin = (–v) = Vin – Vp = (–Vin) – (+Vp)



(1)



vout = (+v) = Vout – Vp = (+Vout) – (+Vp) ∴ v = Vout – Vp (2)





Equate (1) and (2):

Rearrange:



∴ v = Vin + Vp



v = Vin + Vp = Vout – Vp



∴ Vout – Vin = ∆V = 2Vp



Vp



Vin



Figure 2.4.2 Hypothetical 180° hyperbolic

orbit in (a) the planet’s frame

of reference (b) the Sun’s frame



45



2



Explaining

and exploring

the solar system



Hitchhiker’s guide to the solar system



I



nstead of selling burgers, two university students

took summer holiday jobs at the NASA Jet Propulsion

Laboratory and made the next half century of solar

system exploration possible. In 1961, Mike Minovitch

proved that gravity assist was possible, and in 1965

Gary Flandro showed (just in the nick of time) that in

1976, 1977 and 1978 the planets would, by luck, be

suitably aligned for a space probe to hitchhike across

the solar system using gravity assist from Jupiter and

Saturn to explore the outer planets Jupiter, Saturn,

Uranus and Neptune. This ‘grand tour’ would not be

possible again until about 2157. Two of the resulting

missions, Voyager 1 and Voyager 2, provided enormous

advances in planetary science, including the nowfamiliar spectacular images of the outer planets and

their moons. Voyager 1 is now at the boundary between

the solar system and interstellar space.



Voyager 1

Saturn

12 Nov 80



Jupiter

9 July 79



Saturn

26 Aug 81



Jupiter

5 Mar 79



Earth 5 Sept 77

20 Aug 77



Pluto

Aug 89



Uranus

27 Jan 86



Voyager 2

Neptune

01 Sept 89



Figure 2.4.3 Voyager 1 and Voyager 2 take the grand tour of the solar

system using gravity assist.



Checkpoint 2.4

1

2

3

4



In a gravity-assist manoeuvre, a probe increases its momentum. Explain how momentum is conserved and why

this might not be obvious to an observer.

During a slingshot manoeuvre, what is the shape of the probe’s orbit in the planet’s reference frame?

What can we say about the probe’s speed (far from the planet), before and after the slingshot manoeuvre, when

viewed from the planet’s frame?

If VP is the orbital speed of the planet, what is the maximum possible speed increase for a probe executing

gravity assist?



2.5 I’m back! Re-entry

It is time to come home. You made it through the launch safely so you’re feeling

lucky. However, you still have another dangerous hurdle to jump—coming back

to Earth or re-entry.



Orbital decay

Account for the orbital decay

of satellites in low Earth orbit.



46



Craft in low Earth orbits do experience some drag or air resistance, because the

atmosphere gradually thins out with altitude in a very roughly exponential trend

(Figure 2.5.1). At altitudes above ~1000 km, drag is considered negligible. Most

Earth-sensing satellites orbit below this altitude, and so have a limited lifetime.

Drag converts orbital kinetic energy into thermal energy, causing the orbital

radius to decrease (orbital decay). The lower the orbit, the greater the air density

(and drag) and so the faster the orbital decay. Sustained orbits are not possible at



space

100



Density (kg m–3)



altitudes below ~160 km (sub-orbital). Satellites in low

orbits can counteract decay by firing rockets from time

to time, but only until the propellant runs out.

Eventually, a craft in a decaying orbit spirals down

towards the Earth’s surface, to burn up catastrophically

like a meteor due to the thermal energy produced by

the enormous drag (see in2 Physics @ Preliminary

p 57). Drag can be enhanced by solar activity such as

temporary increases in the Sun’s output of ultraviolet

radiation, which inflates the upper atmosphere,

making the orbital decay rate unpredictable. Normally

a satellite will burn up completely in the atmosphere,

although parts of larger craft (such as the Russian Mir

Space Station or the US Skylab) occasionally make it to

Earth’s surface.



10–5



10–10



10–15



0



100



200



300



400 500 600

Altitude (km)



700



800



900 1000



Figure 2.5.1 Graph of typical air density in the Earth’s atmosphere

versus altitude



Rest In Pieces



I



n the late 70s, NASA’s first space station, Skylab, was

crippled with low propellant and damaged gyroscopes.

Enhanced solar activity increased its orbital decay and the first

launch of the newly developed Space Shuttle was delayed,

preventing repair missions. So, in July 1979, it fell on Australia.

Pieces too large to burn up on re-entry landed in a line between

Esperance and Rawlinna, Western Australia. Many pieces are in

the Esperance Museum, but the largest (Figure 2.5.2) is now in

the United States Space & Rocket Center. The Shire of Esperance

sent the US government a $400 fine for littering. The fine was

finally paid in April 2009 in a Californian radio publicity stunt,

using listeners’ donations.



of Skylab

Figure 2.5.2 The largest surviving fragment



Safe re-entry corridor

If a spacecraft is carrying a human crew, or if the craft needs to be retrieved, then

plunging into a fiery re-entry like a meteorite is not an option. A safer return

from low Earth orbit normally starts by retro-firing rockets to slow the craft

down so it begins to fall into a lower energy, lower altitude orbit. There the

higher air density starts to slow the craft further. At the bottom of LEO, orbital

speed is nearly 7.8 km s–1. Orbiting spacecraft could not carry enough propellant

to ease down to the surface. The crew has no choice but to head bravely towards

the Earth at the correct angle, using only high technology and clever physics to

protect them.

Safe re-entry is a balance between two forces: drag and lift. Drag is the

deceleration force; lift is the force that keeps an aeroplane in the air. Air moving

relative to the craft creates pressure differences. If pressure underneath is greater

than that above, lift results. The shape and orientation of the craft and the

re-entry angle all affect the ratio of these two forces.



Discuss issues associated with

safe re-entry into the Earth’s

atmosphere and landing on the

Earth’s surface.

Identify that there is an

optimum angle for safe re-entry

for a manned spacecraft into

the Earth’s atmosphere and the

consequences of failing to

achieve this angle.



47



2



Explaining

and exploring

the solar system



s

in



u



cie

ffi



nt d



rag



re-entry corridor

sive drag

c

ex es



Figure 2.5.3 The safe re-entry corridor



The three main issues behind safe re-entry are:

1 minimising the effects of deceleration (g-force)

2 managing the effects of heating

3 landing the craft safely in the right place.

The first two issues lead to the existence of a narrow range of safe re-entry

angles and speeds (Figure 2.5.3). Drag is both good and bad. Drag provides the

spacecraft with brakes, but it also produces the copious amounts of thermal

energy that could destroy the craft. If the approach into the atmosphere is at too

shallow an angle, drag will be too small, the air flow will provide too much lift

and the craft will skip over the atmosphere instead of entering. If the angle is

too steep, drag will be too large, producing excessive heat and deceleration g-force,

which would destroy the craft and crew.



Deceleration

As at launch, astronauts’ seats during re-entry are oriented perpendicular to, and

facing (‘eyeballs in’) the direction of acceleration, but this time acceleration is

opposite to velocity, so they look backwards.

Traditional re-entry vehicles (such as were used in the 1960s and 1970s) were

teardrop-shaped capsules with the blunt end pointing forward (Figure 2.5.4).

They allowed very little or no control once re-entry had begun and provided very

little lift. Such a re-entry is called ballistic re-entry and requires larger re-entry

angles. This kind of capsule subjected the astronauts to a maximum re-entry

g-force of anywhere between 6 and 12. The Apollo re-entry angle was between

5.2° and 7.2°

The Space Shuttle introduced in 1981 has wings that provide lift and flightcontrol structures (such as elevons, a rudder/speed brake and a body flap) that

allow considerable control over the descent, adjusting the vehicle’s aerodynamics

to the changing density of the air, and making re-entry more gentle, with a

maximum g-force of about 2–3. This degree of control also widens the safe re-entry

corridor, allowing a gentle, low-g 1–2° re-entry. This is called glide re-entry.

To further decrease descent speed without excessive g-force, the Shuttle performs

a series of S‑shaped turns by rolling and banking, gently enhancing drag.

The Russian Soyuz capsule, in use continuously (in modified form) since the

1960s, is a more spherical variation of the traditional capsule shape but with

attitude control thrusters, which provide some glide control during re-entry.

It usually yields a g-force of 4–5, but sometimes up to about 8 for a completely

ballistic re-entry.

Ballistic re-entries are high acceleration but quick—between 10 and 15

minutes. A full glide re-entry is low acceleration but slow—Shuttle re-entry,

for example, takes about 45 minutes. Soyuz is intermediate and takes about

30 minutes.

Heating

On re-entry, vehicles travel at well above the speed of sound. The speed of sound

is sometimes called Mach 1 (after Ernst Mach (1838–1916) a physicist and

philosopher who studied gas dynamics). Twice the speed of sound is called

Mach 2 and so on. Supersonic means travelling faster than Mach 1. Hypersonic

usually means faster than Mach 5 (oversimplifying somewhat).

Pressure builds up in front of projectiles. Sudden pressure changes normally

propagate away as sound (at the speed of sound). In supersonic flight, however,

48



space

this pressure wave is too slow to move out of the projectile’s way, so the pressure

builds up to very high levels, forming a shock wave—the air equivalent of the

bow wave in front of a speed boat.

The enormous mechanical energy of orbit must go somewhere. Drag

converts it to thermal energy. Contrary to common sense, in hypersonic flight,

a blunt projectile with more drag actually gets less hot than a more streamlined

one. In the 1950s, Harvey Julian Allen proved this theoretically and explained

why the sharp nose cones of intercontinental ballistic missiles were vaporising on

re-entry. Hypersonic wind tunnel tests (see Figure 2.5.5) confirmed his theory.

In hypersonic flight, much of the heat generation takes place in the shock

wave (the dark line wrapping around the front of both projectiles in Figure 2.5.5).

The shock wave does not touch the blunt projectile (Figure 2.5.5a) and so

doesn’t transfer the heat efficiently to it, but it does touch the tip of the sharp

projectile (Figure 2.5.5b), which gets much hotter. For this reason, re-entry

vehicles (including the Space Shuttle) are blunt at the front, and hence the

traditional teardrop shape of capsules.

The blunt front of the vehicle is also coated with a suitable heat shield with

very high melting and vaporisation temperatures. It is also highly insulating to

slow the rate of thermal conduction. Thermal insulating materials in most

applications are almost always very porous because tiny pockets of gas are very

poor thermal conductors. Some insulator materials are also designed to be highly

light-absorbing (black) in the visible and near infra-red parts of the spectrum

because such surfaces, when hot, also radiate thermal energy away more

efficiently (radiative cooling).

Tiles on the Shuttle surface are made of 90% porous silica fibre, which is an

excellent high melting point insulator, but it is brittle. The tiles on the hottest

parts (the underside and leading edges) are also coated with a tough black glass

to enhance radiation of thermal energy, but also to provide mechanical strength.

Broken tiles were believed to be responsible for the destruction during re-entry

of the Shuttle Discovery in 2003.

In more traditional space capsule ballistic re-entry, drag is higher, so heat is

generated more rapidly, and insulation and radiation alone are not enough. In

these cases, the insulating heat shield is also designed to vaporise and erode

(ablation). The hot, vaporised and ablated material carries thermal energy away

rather than conduct it to the capsule, similar to the way in which evaporating

sweat carries away excess heat from your skin. The pressure from this ablation

also helps to push away the hot gas convecting from the shock wave. The shield

must be thick enough to last the journey and provide sufficient insulation.

Two modern examples of ablating materials are phenolic impregnated carbon

ablator (PICA) and silicone impregnated reusable ceramic ablator (SIRCA). In

the Chinese space program, one of the ablation materials used is blocks of oak

wood. It’s cheap and easy to work. As it chars, it forms charcoal, which is porous

and almost pure carbon, making it an extraordinarily good thermal insulator

with a very high melting point. Another advantage is that porous carbon is very

black and radiates thermal energy efficiently. However, it is mechanically weaker

than more ‘high-tech’ ablation materials.

During re-entry, superheated air surrounding the vehicle is ionised. The air

becomes a plasma—a conductive soup of free positive and negative charges that,

like the Earth’s ionosphere (see in2 Physics @ Preliminary pp 153–4), reflects



a



b



c



d



Figure 2.5.4 Re-entry vehicles: (a) Gemini

1964–1966 (b) Apollo 1966–1975

(c) Soyuz 1960–present

(d) Space Shuttle 1981–present

49



2



Explaining

and exploring

the solar system

a



b



Figure 2.5.5 Hypersonic wind tunnel tests. (a) The crescent-shaped shock wave is detached from

the blunt projectile, but (b) touches the tip of the sharp projectile.



radio waves, so the astronauts cannot communicate with the Earth for several

minutes during re-entry. This problem has been solved for the Shuttle by

communicating via a satellite above it, since only the bottom of the Shuttle has

significant ionisation.



Landing

Drag depends on the projectile’s cross-sectional area and speed. Drag cannot stop

a projectile completely because, during deceleration, drag decreases until it

exactly cancels the weight of the projectile and deceleration stops—the projectile

has reached terminal speed (see in2 Physics @ Preliminary p 45). The terminal

speed of a capsule is too high for it to land safely. To slow the capsule further for

the landing, drag is enhanced (and terminal speed decreased) by using parachutes

to increase the effective area of the capsule.

The final ‘touchdown’ could be on land (typical of Russian missions) or a

‘splashdown’ in the water (typical of US missions pre-Shuttle). Russian Soyuz

also has soft-landing engines that fire just before it touches the ground.

The Space Shuttle lands on a runway, much like an aeroplane (Figure 2.5.6)

but it uses parachutes to help it brake. During landing, the Shuttle (which has

been described as being ‘like flying a brick with wings’) is controlled entirely by

computer.

Another issue is accurate targeting of the landing site. The steeper the

re-entry angle, the smaller the horizontal component of motion (range) and so

the more accurate the prediction of the final landing site. However, the Shuttle

makes up for its shallow re-entry, because its aeroplane-like flight-control

structures allow adjustment of the landing path. The shape of the landing path is

also designed to be more forgiving. The Shuttle approaches the runway roughly

opposite to the landing direction. Four minutes from touchdown, it does a

‘heading-alignment’ loop, to adjust precisely to the direction of the runway

(Figure 2.5.6).



50



space



Altitude 25 000 m



Mojave

Runw

ay 23

E

d

Airfor wards

ce Ba

se



Figure 2.5.6 Scale drawing of the relatively gentle descent of the Space Shuttle. The Shuttle is

drawn at 1 minute intervals to touchdown. The squares on the ground are 10 nautical

miles (18.5 km) wide.



Checkpoint 2.5

1

2

3

4

5

6

7

8

9

10

11



Define orbital decay and explain what causes it.

Because of drag, satellites at altitudes below ~1000 km can do nothing to combat orbital decay. True or False?

Explain.

What other astronomical body can affect the rate of orbital decay? Explain.

Discuss how drag is ‘good and bad’ for re-entry.

Outline what can happen if a spacecraft attempts re-entry with too shallow or too steep an angle.

Explain why astronauts face backwards during re-entry, unlike at launch.

Outline why occupants of the Space Shuttle experience lower g-force during re-entry than in the more traditional

re-entry vehicles.

Define the terms supersonic and hypersonic.

What is a shock wave?

Outline why a pointy hypersonic projectile is more likely to melt than a blunt one.

Explain why a capsule with a parachute slows down more than without one.



51



2



Explaining

and exploring

the solar system



PRACTICAL EXPERIENCES

CHAPTER 2



This is a starting point to get you thinking about the mandatory practical

experiences outlined in the syllabus. For detailed instructions and advice, use

in2 Physics @ HSC Activity Manual.



Activity 2.1: Development of space exploration

Identify data sources, gather,

analyse and present

information on the contribution

of one of the following to the

development of space

exploration: Tsiolkovsky,

Oberth, Goddard, EsnaultPelterie, O’Neill or von Braun.



Use the template provided in the activity manual to extract information about

your chosen scientist. Process this information to make a short oral presentation

to the class.

Discussion questions

1 For the scientist that you have researched, list their main contributions

to space exploration.

2 Explain how later scientists have benefited from this research.

Extension

3 Werner von Braun’s great Russian rival, the ‘Chief Designer’ for the USSR

space program Sergey Korolyov, is not as familiar as some of the names

mentioned in the syllabus, despite leading the launch of the first artificial

satellite, Sputnik, in 1957. This is probably because his name was kept

secret by the communist government of the USSR until after his death in

1966. You may also want to research his contribution to space exploration.



Activity 2.2: Uniform circular motion

Solve problems and analyse

information to calculate the

centripetal force acting on

a satellite undergoing uniform

circular motion about the

Earth using:





F =m



v2

r



Perform an experiment that will allow you to determine the relationship between

the radius of a satellite’s orbit around the Earth and its gravitational force.

Equipment: string, rubber stopper, mass carrier and masses, electronic scales,

glass or plastic tube, paperclip, sticky tape, metre ruler, stopwatch.

Discussion questions

1 From your experimental data,

determine the mathematical

relationship between the

orbital radius of a satellite and

its tangential velocity for a

given centripetal force.

2 Describe the method you

would use to determine the

centripetal force on a small

model satellite.



tension

glass or

plastic tube

paperclip



string



mass carrier



Figure 2.6.1 Force on mass moving in a

horizontal circle



52



mg



Chapter summary

•

•

•

•

•

•

•

•



•



•

•



•



•

•

•



•



•



Rocket thrust is the reaction to the force exerted on the

exhaust exiting the nozzle.

For a fixed mass of exhaust per unit time, thrust

increases as exhaust speed increases.

As the mass of remaining propellant decreases, a rocket’s

acceleration (and g-force) increases.

There are two kinds of rocket engines: liquid and solid

propelled.

By jettisoning used stages, a rocket’s mass is decreased,

allowing more payload to be carried.

g-force is the ‘apparent weight’ experienced during

acceleration, divided by true weight on Earth.

a

a

Vertical g-force = v + 1. Horizontal g-force = h .

g

g

To increase g-force tolerance, astronauts are seated with

bodies horizontal, and looking in the direction of the

acceleration (‘eyeballs in’) for both lift-off and re-entry.

A spacecraft launched eastwards has the extra initial

velocity of Earth’s rotation. This is greatest at the

equator (465 m s–1).

Launches in the direction of Earth’s orbital velocity

obtain an initial velocity boost of 3.0 × 104 m s–1.

Gravity provides the centripetal force for satellite orbits:



mv 2

(for circular orbits)

R

Kepler’s laws (apply to any two bodies if the central

body has a very much larger mass):

1 Orbits of planets are ellipses, with the Sun at one

focus.

2 Planets sweep out equal areas in equal times.

T2

3 Law of periods:

= a constant

a3

a 3 GM

Explicit form of Kepler’s third law: 

=

T 2 4π2

Eccentricity is a measure of the elongation of an ellipse.

A circle is an ellipse of zero eccentricity.

Periapsis is the position of closest approach to the

central mass and fastest orbital speed (perihelion for the

Sun, perigee for Earth).

Apoapsis is the position of furthest distance from the

central mass and slowest orbital speed (aphelion for

the Sun, apogee for Earth).

If the mass of a satellite is not negligible when compared

to that of the central body, then both masses orbit with

the same period around the system’s centre of mass.



•

•





•

•

•

•



•

•

•



Fc =



•



•



•

•



•



•

•

•



space



GM

r

Two-body orbital mechanical energy

1

GmM :

ME = mv 2 −

2

r

Magnitude of orbital speed: v =



– ME < 0 (bound), orbit is closed or stable (circle

or ellipse); velocity < escape velocity

– ME = 0 (borderline unbound) orbit is parabolic;

velocity = escape velocity

– ME > 0 (unbound), orbit is hyperbolic;

velocity > escape velocity

GmM

.

For stable orbits, ME = −

2a

Orbits are symmetrical in shape and speed.

Low Earth orbit (LEO): altitude between ~160 and

~2000 km.

At altitudes below ~1000 km, drag causes orbital decay.

Upper atmosphere can be inflated by increased solar UV

radiation, increasing drag.

Orbits below ~1000 km are protected from the Van

Allen radiation belts by the atmosphere and distance.

Geosynchronous orbit: T = 1 sidereal day

(23 h 56 m 4 s).

Circular geosynchronous orbits over the equator are

called geostationary. These orbits are used extensively

for communication satellites (r = 42 164 km).

A space probe entering a temporary hyperbolic orbit

behind an orbiting planet can gain momentum via

gravity assist or the slingshot effect.

For gravity assist, the maximum possible change in

speed of the probe in the Sun’s frame of reference is

twice the planet’s orbital speed Vp.

Drag converts orbital KE into thermal energy.

Safe re-entry angle: if angle is too low, the craft will skip

off atmosphere; if angle is too high, g-force and heating

rate are too large.

Much heating takes place in the hypersonic shock wave.

Blunt-fronted re-entry vehicles are used because the

shock wave is detached from the craft.

Heating of a spacecraft on re-entry is reduced by an

insulating and radiating heat shield.

Traditional capsules also use ablation of the heat shield

to dissipate heat.

Parachutes decrease terminal velocity by increasing the

effective cross-sectional area.



53



2



Review questions



Explaining

and exploring

the solar system



Physically speaking

For each type of orbit, fill in the missing information. One has been done already.



Name of orbit



Sign of two-body ME



Open or closed



v >, =, < vescape



Bound or unbound



Negative



Closed



<



Bound



Geostationary

Slingshot (in planet’s frame)

Elliptical

Hohmann (see Physics Focus)

Parabolic

Hyperbolic

Halley’s Comet

Circular

Molniya



Reviewing

Solve problems and analyse

information to calculate the

centripetal force acting on a

satellite undergoing uniform

circular motion about the

Earth using:

2



mv

r

Solve problems and analyse

information using:

mm

F =G 1 2

d2

Analyse the forces involved

in uniform circular motion

for a range of objects,

including satellites orbiting

the Earth.

F =



1 What was the first solid rocket propellant and who invented it?

2 Assuming that propellant is burned at a constant mass per unit time, use the

equation for thrust to explain why forcing exhaust gas through a narrow nozzle

increases thrust.



3 List the advantages of both solid and liquid propellants.

4 Discuss why the vertical g-force formula has a ‘+ 1’ term, but the horizontal

formula doesn’t.



5 Describe a situation during launch in which astronauts would experience a g-force

greater than zero but less than 1.



6 At the bottom of a bungee jump with the cord attached to the ankles, one can

easily experience a g-force of 3, the maximum normally allowed for Shuttle

launches. Describe three important differences in the way the g-force is

experienced in these two situations.



7 Explain why launch facilities are usually built as close to the equator as

is practical.



8 Describe the circumstances under which a star would not sit at the focus of

a planet’s elliptical orbit.



9 Discuss why we only briefly see Halley’s Comet with an unaided eye every 76 years,

even though it is in orbit continuously around the Sun.



10 Outline what happens to the period of a satellite if its semimajor axis is reduced

by a factor of 4.



11 A space probe approaches a planet in a hyperbolic orbit. Discuss the condition

that must be fulfilled to move it into a stable orbit of the planet and describe how

it might be achieved.



12 By re-examining the gravity assist worked example on page 45, show that the

magnitude of change in the probe’s velocity in the Sun’s reference frame is twice

the probe’s initial speed in the planet’s reference frame (that is, |Vout – Vin| = 2v).



13 A satellite is in a highly elliptical orbit around Earth such that, at perigee, it is

briefly at an altitude of less than 1000 km. Over many orbits, the altitude at

apogee decreases (the orbit becoming more circular). Explain why this occurs.



14 List two reasons why (human-crewed) space stations are always in low Earth orbit.

54



space



Solving Problems

15 Calculate the two-body gravitational potential energy for a system consisting



Analyse the forces involved in

uniform circular motion for a

range of objects, including

satellites orbiting the Earth.



of a 1.00 kg test mass sitting on the surface of the Earth. How far would

the test mass need to be from the Sun so that the two-body GPE of the

test mass–Sun system is the same value? Estimate roughly where that

position would be in relation to the orbital radii of the planets.



16 Typically, at launch, the Shuttle’s main engines, with an effective exhaust

velocity of 4460 m s–1, produce a thrust of 5.45 × 106 N. The two solidfuel rocket boosters, with an effective exhaust velocity of 2640 m s–1,

produce 1.250 × 107 N each.

a Calculate the combined rate (in kg s–1) at which propellant is used

at launch.

b Assuming a mass at launch of 2.03 × 106 kg, calculate the Shuttle’s

acceleration at launch and 1 minute later, assuming the above

specifications remain constant. (Hint: Don’t forget gravity.)



17 On a roller-coaster, you round the top of a circular hump in the track of







5.00 m radius. You have a g-force meter with you and at the moment

you’re at the top it reads a vertical g-force of 0.00.

a What is your weight at that moment?

b What is the magnitude of the normal force exerted on you by the seat

at that moment?

c What is your centripetal acceleration?

d Calculate your speed at the top.

e Assuming friction and air resistance are negligible, calculate your

horizontal g-force at that moment.



18 Prunella spins a weight (mass m) on a string (length L) in a horizontal



Solve problems and analyse

information to calculate the

centripetal force acting on a

satellite undergoing uniform

circular motion about the Earth

using:



circle (Figure 2.6.2) to illustrate the relationship between orbital speed

and centripetal force for an orbiting satellite. Renfrew says: ‘Because of

the weight, the string isn’t horizontal so the orbital radius is R = L sin θ,

and the centripetal force is Fc = T sin θ.’



Prunella then says: ‘Yeah, but as long as the orbital speed v is high







enough, θ will be very close to 90° so you can use the approximation that

string tension T is the centripetal force and the string length L represents

orbital radius R.’

v2

Show that as long as orbital speed v fulfils the condition

> 7g,

R

then L is no more than 1% larger than the true orbital

radius R and T is no more than 1% larger than the true

centripetal force Fc.



Solve problems and analyse

information using:

mm

F =G 1 2

d2







mv 2

r

Analyse the forces involved in

uniform circular motion for a

range of objects, including

satellites orbiting the Earth.

F =





T sin θ

θ



L



mg



T

θ



R



Figure 2.6.2 Spinning weight model

of a satellite



55