Download Chapter 13 Exploring the final frontier

CHAPTER 13 EXPLORING THE FINAL FRONTIER
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QUESTIONS
13.1 Cosmic forces
1.
Given that the Venus year is equal to 0.615 Earth years, determine its distance from the
Sun in AU.
2.
Using the same process, complete the following table.
3.
Determine the gravitational force of attraction between Jupiter (1.90 × 1027 kg) and its
moon Ganymede (1.48 × 1023 kg). The orbital radius of Ganymede is 107 million km.
4.
(a)
Calculate the gravitational force between Ganymede and a 250 g apple lying on
Ganymede’s surface. The radius of Ganymede is 2631 km.
(b)
This force is the weight of the apple on Ganymede. Can you deduce the value of g
at that location?
5.
Calculate the value of g at the surface of Venus. The mass of Venus is 4.869 × 1024 kg
and its radius is 6052 km.
6.
Determine the weight of a 75 kg astronaut when she is in orbit around Saturn’s largest
moon, Titan, at an altitude of 150 km. Titan’s mass is 1.35 × 1023 kg and its radius is
2575 km.
13.2 Orbits
7.
8.
A ride at an amusement park whirls the rider around on the inside surface of a large
cylinder of radius 5.0 m. It takes 2.5 s to complete a circle. Calculate:
(a)
the centripetal force acting on a 90 kg rider
(b)
what multiple of his normal weight this figure represents. (This, of course, is the gforce experienced.)
Determine the centripetal acceleration of an aeroplane executing a turn of radius 500 m at
a speed of 230 km h–1.
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9.
Calculate the orbital velocity required by the Space Shuttle when at an orbital altitude of
250 km above the surface of the Earth. The mass of the Earth is 5.97 × 1024 kg and its
radius is 6380 km.
10. Sample problem 13.4 referred to Apollo command modules orbiting the Moon at an
altitude of 110 km. Calculate the orbital velocity required to do this. The mass of the
Moon is 7.35 × 1022 kg and its radius is 1738 km.
11. Determine the orbital velocity of a GPS satellite orbiting at an altitude of 20 000 km
above the Earth. Can you work out from this how long it takes for the satellite to
complete an orbit?
13.3 Rockets
12. Another model rocket is packed with 10.5 g of fuel, bringing its pre-launch mass to
88.5 g. This fuel will deliver a thrust of 3.95 N for a period of 2.3 s. Assuming a vertical
launch, determine the rocket’s initial and final acceleration and g-force.
13.5 Relativity
13. For the scenario described in sample problem 13.10, calculate the length of the sneeze as
seen by the person on the platform if the train is travelling at:
(a)
0.1c
(b)
0.9c
(c)
0.99c.
14. For the scenario described in sample problem 13.12, calculate the length of a carriage as
seen by a person on the platform if the train is travelling at:
(a)
0.1c
(b)
0.9c
(c)
0.99c.
15. If Superman’s rest mass is 100.0 kg (lots of steely muscle), calculate his mass if he is
travelling at:
(a)
0.1c
(b)
0.9c
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(c)
0.99c.
16. Describe what happens to the mass of an object as its velocity approaches the speed of
light, c.
Review questions
Understanding
1.
What would happen to the strength of the force of attraction between two masses if
the distance between them was to halve while the masses themselves were each to
double?
2.
Draw the gravitational field:
(a)
as it exists within the room in which you are currently sitting
(b)
as it exists around the planet Mars.
3.
At what points during a roller-coaster ride does a rider experience an increase in g-force
as an astronaut does during lift-off?
4.
Draw a diagram of a satellite orbiting the Earth, showing all forces acting on the satellite.
5.
Compare and contrast, in words only, low Earth orbits and geostationary orbits.
6.
(a)
Describe the slingshot effect of a planetary swing-by.
(b)
What benefit is achieved by this manoeuvre?
7.
The furthest that any human has travelled is the Moon, and this only took place over a fouryear period from 1968 to 1972. Describe some of the design considerations for a spacecraft
that must carry live passengers.
8.
How would a ‘fly-back’ booster rocket reduce launch costs?
9.
(a)
Describe the function of an ion propulsion drive.
(b)
What advantages does this type of engine have over a conventional rocket engine?
10. How could you tell if your spacecraft, far from any planet or star, is standing still or
travelling with a uniform velocity? What is the name given to this idea?
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11. In classical physics, time was assumed to be absolute and the maximum velocity
of interaction infinite. How are these same concepts viewed in the theory of relativity?
12. In one paragraph, describe what is meant by the ‘relativity of simultaneity’.
13. In terms of the energy required, accelerating a spacecraft to light speed is an
impossibility. Explain why this is so.
Application
14. Given the following data, determine the magnitude of the gravitational attraction
between:
(a)
Jupiter and its moon Callisto
(b)
Jupiter and the Sun.
Mass of Jupiter = 1.9 × 1027 kg
Mass of Callisto = 1.1 × 1023 kg
Mass of the Sun = 1.99 × 1030 kg
Distance between Jupiter and Callisto = 1.88 × 109 m on average
Distance between Jupiter and the Sun = 7.78 × 1011 m on average
15. Determine the magnitude of the force of gravitation between a book of mass 1 kg and a
pen of mass 50 g lying just 15 cm from the centre of mass of the book.
16. Complete the following table to determine the acceleration due to gravity and weight
force experienced by an 80 kg person standing on the surface of each of the planets or
moons indicated.
17. A certain model rocket has a pre-launch mass of 87.3 g, of which 10.5 g is propellant. It
is able to deliver a thrust of 6.1 N. Assuming that the rocket is fired directly up,
determine:
(a)
the initial rate of acceleration and initial g-force
(b)
the final rate of acceleration and final g-force just prior to exhaustion of the fuel.
18. A 400 g rock is tied to the end of a 2 m long string and whirled until it has a speed of
12.5 m s–1. Calculate the centripetal force and acceleration experienced by the rock.
19. Calculate the time that light emitted by the Sun would take to reach each of the planets of
the solar system, using the table at the top of the following page. (Recall that the speed of
light is 3 × 108 m s–1.) In addition, determine the time that a spacecraft would take to cover
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the same distance travelling at 100 000 km h–1.
20. (a)
(b)
It has been suggested that a light sail would be able to reach Proxima Centauri in
just 40 years. What average velocity in m s–1 would be required to achieve this?
The distance to Proxima Centauri is 4.3 light-years.
What fraction of the speed of light does this velocity represent?
21. As the Moon orbits the Earth it has an orbital speed of approximately 3660 km h–1. Its
proper diameter is 3480 km, so what do we observe as its diameter?
22. A muon is a subatomic particle with an average lifetime of 2.2 microseconds when
stationary. In a burst of cosmic rays in the upper atmosphere, muons are observed to have
a lifetime of 16 microseconds. Calculate their speed.
Challenge
23. Draw a graph that shows the value of g at every 5000 km above the surface of the Earth
up to an altitude of 40 000 km. The mass of the Earth is 5.97 × 1024 kg and the radius of
the Earth is 6378 km at the equator.
24. If a rocket had a fuelled mass of 32 000 kg, of which 85% was fuel, and a thrust of
400 000 N, determine:
(a)
the rate of acceleration and g-force at lift-off
(b)
the rate of acceleration and g-force just prior to exhaustion of the fuel, assuming it
is now travelling horizontally and accelerating up to orbital velocity.
25. A person hurls a rock of mass 0.25 kg horizontally from the top of Mt Everest, at an
altitude of 8.8 km above sea level.
(a)
What velocity would be required by the rock so that it would orbit the Earth and
return to the thrower, who is still waiting on Mt Everest? The mass of the Earth is
5.97 × 1024 kg and its mean radius is 6380 km.
(b)
How long would the thrower have to wait for the rock’s return?
(c)
What is the magnitude and direction of the centripetal force acting on the rock?
26. Apollo rockets to the Moon would always begin their mission by launching into a low
orbit with an altitude of just 180 km. An orbit of this height can decay quite rapidly, but
they were never there for very long before boosting out of orbit towards the Moon. For
their orbit, calculate:
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(a)
the required velocity in km h–1
(b)
the period of the orbit
(c)
the acceleration of the spacecraft
(d)
the centripetal force acting on the spacecraft.
Use 110 000 kg as the mass of the spacecraft in orbit. Additional data can be found in the
previous question.
27. Use Kepler’s law of periods to determine the time required for the following planets to
complete an orbit of the Sun. The radius of the Earth’s orbit is 150 × 106 km.
28. Jake the flying ace needs to hide his new experimental high-velocity plane in the hangar,
away from the view of foreign spy satellites. ‘You’ll ne’er do it,’ says Jock, the Scottish
flight engineer, ‘The hangar’s barely 80 m long but yerr plane there is over 120.’
‘She’ll be right, mate,’ replies Jake, ‘Its just a matter of going fast enough!’ Jock
operates the hangar door while he watches Jake approach in his plane.
(a)
Jock sees the plane contract as it speeds up. How fast must it be travelling for him
to be able to quickly close the door with the plane, at least momentarily, contained
inside?
(b)
At the speed you calculated in part (a), how long is the hangar as seen by Jake in
the approaching plane? Will the plane fit in the hangar as judged by Jake?
(c)
How is it possible for Jake and Jock to perceive the situation so differently?
29. A super rocket racer has a proper length of 30 m and can fly at 0.3c. Calculate:
(a)
the length of the aircraft at speed when observed from the Earth
(b)
the time difference between the clocks of the pilot and his airbase if they were
perfectly synchronised prior to lift-off and the racer was aloft and at speed for
10 hours
(c)
the mass of the speeding aircraft when observed from Earth.
30. A Martian spaceship travelling at near light speed passes a stationary Venusian spaceship.
Each is made of glass, and in each the occupants are holding a dance party.
(a)
Describe the dancing Venusians as seen by the Martians.
(b)
Describe the dancing Martians as seen by the Venusians.
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(c)
Are your answers to (a) and (b) contradictory? Explain.
31. Calculate the travel time to the following destinations travelling at 0.999c.
(a)
Pluto when at a distance of 6.1836 × 10–4 light-years from the Earth
(b)
Proxima Centauri at a distance of 4.2239 light-years
(c)
Sirius at a distance of 8.6433 light-years
(d)
Alpha Crucis at a distance of 521.95 light-years
(e)
The Andromeda galaxy at a distance of 2.53 × 106 light-years
Notes:
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