Physics with KSP 1

The Kerbal Math & Physics Lab / Chapter 1
1. Algebra
Gravity and Acceleration
Acceleration is any change in motion, including a change in direction, or an increase or decrease
in speed. Isaac Newton’s first law of motion states that a force is required to cause an object to
accelerate. Newton’s second law of motion, described by the equation F = ma, states that the
magnitude of force, F, that causes a change in the motion of an object is equal to the mass, m,
of the object, times the acceleration, a, of that object.
Newton’s law of gravitation, expressed by the equation F =
𝐺𝑀𝑚
𝑟2
, describes how the force of
gravity is related to the mass of two objects and the distance between them. In this equation,
G, the constant of proportionality, is called the universal gravitational constant. The values M
and m are the masses of two objects experiencing the force of gravity and r is the distance
between the center of mass of each of the objects. From the law of gravitation, we say the
force of gravity is proportional to the product of the masses and inversely proportional to the
square of the distance between them.
To begin, we consider how gravity causes objects to move by calculating the acceleration due to
gravity on Earth and on the planet Kerbin.
1. Use the equations F = ma and F =
𝐺𝑀𝑚
𝑟2
to solve algebraically for a single formula for a,
the acceleration of an object due to gravity based only on the values G, M and r.
In real-life, and in the game Kerbal Space Program (KSP), the value G = 6.67 x 10-11 m3/(kg*s2) is
used as the universal gravitational constant.
2. The mass of the Earth is M = 5.97 x 1024 kg and its radius is approximately r = 6,378,000
meters. Compute the acceleration due to gravity at the surface of the Earth.
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The Kerbal Math & Physics Lab / Chapter 1
3. In KSP, the planet Kerbin has a mass of M = 5.29 x 1022 kg and a radius of r = 600,000
meters. Compute the acceleration due to gravity at the surface of Kerbin. Compare
with the acceleration due to gravity at the surface of Earth.
4. Density is mass divided by volume. Compute the density of the Earth and the density of
Kerbin. Which has higher density?
5. Search online to find the density of lead. Write the density of lead at standard
temperature and pressure. What is the name and density of the element on the
periodic table with the highest density? (Note: this isn’t the element with the highest
atomic mass, density is not usually included on the periodic table.) How does this
element’s density compare with the density of Kerbin?
6. What is the acceleration due to gravity on an object in orbit at an altitude of 100 km
above Kerbin?
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The Kerbal Math & Physics Lab / Chapter 1
As we have seen, acceleration due to gravity can be computed with the equation 𝑎 =
𝐺𝑀
𝑟2
.
This means acceleration due to gravity changes with the values of the mass M and the distance
r. We define weight as the force due to gravity, and compute weight using the equation
𝐺𝑀
𝐹 = 𝑚𝑎 = 𝑚 ( 𝑟 2 ) =
𝐺𝑀𝑚
𝑟2
.
As stated previously, the value G is a constant of proportionality and is determined by the
choice of units. If we measure mass in kilograms, distance in meters, and time in seconds, we
use G = 6.67 x 10-11 m3/(kg*s2) and with these units, weight is measured in Newtons.
If weight is measured in pounds, this is also a measure of the force of gravity. Stating an
object’s weight in pounds is not directly a measure of its mass because weight changes with
changes in the gravity field, that is, it depends on the values of M and r.
To convert units we can use the following: 1 pound = 4.45 Newtons of force (anywhere in the
universe including anywhere in the Kerbal system). Near the surface of the Earth and Kerbin,
any object with a mass of 1 kilogram has a weight of 2.2 pounds. Note this means, at the
surface of the Earth or Kerbin, a 1 kilogram mass has a weight of F = 2.2 pounds x 4.45
Newtons/pound = 9.8 Newtons.
We feel weight on the surface of Earth when the ground below our feet provides an equal and
opposite force keeping us from accelerating toward the center of the Earth. We also feel
weight when we lift something and our muscles provide a force that opposes gravity. Imagine
if you and a briefcase were both falling together: You would no longer feel the weight of the
briefcase. Objects in orbit around a planet or moon are also in free-fall, but moving so fast
horizontally to the surface, without an atmosphere to slow them down, that gravity only bends
their path. Astronauts appear weightless because they are in free-fall and the only force acting
on them is gravity. They appear weightless as long as there are no other forces (such as a
rocket’s thrust) acting to oppose gravity and their weight is apparent only in the way gravity
affects their trajectory.
As we have seen, acceleration due to gravity at the surface of the planet Kerbin and at the
surface of Earth is exactly the same.
For Kerbin, with a mass M = 5.29 x 1022 kg and radius r = 600,000 meters,
𝑎=
𝐺𝑀
𝑟2
=
(6.67×10−11 )(5.29×1022 )
6000002
= 9.8 m/s2
and for Earth, with a mass M = 5.97 x 1024 kg and radius r = 6,378,000 meters,
𝑎=
𝐺𝑀
𝑟2
=
(6.67×10−11 )(5.97×1024 )
63780002
= 9.8 m/s2
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The Kerbal Math & Physics Lab / Chapter 1
Because acceleration due to gravity at the surface of Kerbin is the same as it is at the surface of
the Earth, the weight of an object on the surface of Kerbin will be the same as the weight on
the surface of Earth. However, because Kerbin is smaller than Earth, the acceleration due to
gravity at the same altitude above each planet is not equal. Thus, the weight of an object in
space at equal altitudes above each planet also will not be the same.
For example, the weight of a mass of m = 50 kg, at an altitude of 100 kilometers above each
planet is computed as follows:
For Kerbin, at an orbital radius of r = 600,000 + 100,000 = 700,000 meters
𝐹=
𝐺𝑀𝑚
𝑟2
=
(6.67×10−11 )(5.29×1022 )(50)
7000002
≈ 360 Newtons
For Earth, at an orbital radius of r = 6,378,000 + 100,000 = 6,478,000 meters
𝐹=
𝐺𝑀𝑚
𝑟2
=
(6.67×10−11 )(5.97×1024 )(50)
64780002
≈ 474 Newtons
Use the previously introduced unit conversions and the equation 𝐹 =
following:
𝐺𝑀𝑚
𝑟2
to complete the
7. A mass of 1 kilogram has a weight of 9.8 Newtons on the surface of Earth and Kerbin.
Find the weight (in Newtons) of a 3000 kilogram satellite on the surface of either planet.
8. A mass of 1 kilogram weighs about 2.2 pounds on the surface of Earth and Kerbin, what
is the weight (in pounds) of a 3000 kg satellite on the surface of Earth and Kerbin?
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The Kerbal Math & Physics Lab / Chapter 1
9. What is the weight, in Newtons and in pounds, of a 3000 kg satellite in orbit at an
altitude of 100 km above Kerbin? Is the satellite weightless while in orbit?
10. Given that a mass of 1 kg weighs about 2.2 lbs at the surface of Earth, find the mass, in
kilograms, of a person that weighs 160lbs at the surface of Earth.
11. What is the weight, in Newtons, of a 160 lb person on the surface of the Earth?
12. What is the weight, in Newtons, of a 160 lb person in orbit 100km above Earth?
Remember Earth’s radius is about 6378 km.
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The Kerbal Math & Physics Lab / Chapter 1
In the formula 𝐹 =
𝐺𝑀𝑚
𝑟2
, when we use the value G = 6.67 x 10-11 m3/(kg*s2), we assume
distance is measured in meters, and mass is measured in kilograms. We could combine the
𝐾
values GMm into one single constant and write 𝐹 = 𝑟 2 and use any units for F and r. Again,
with this equation, we can say the force of gravity is inversely proportional to the square of the
distance. Given any corresponding values for F and r, we can solve for K and, remaining
consistent in our choice of units, then use that to answer questions such as the following:
13. Suppose an astronaut weighs 160 pounds on the surface of the Earth, which has a
radius of about r = 3960 miles. What is the astronaut’s weight in orbit at an altitude of
100 miles?
14. Suppose Jebidiah Kerman, a Kerbal astronaut, weighs 80 lbs on the surface of Kerbin
which has a radius of 600 km. What is Jeb’s weight at an altitude of 100 km above
Kerbin?
15. The International Space Station (ISS) orbits the Earth at an altitude of about 250 miles
above the surface. Earth’s radius is about 3960 miles. Calculate your own weight, in
pounds, if you were onboard the ISS. Derive a formula that you could use to find
anyone’s weight, in pounds, if they were onboard the ISS.
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The Kerbal Math & Physics Lab / Chapter 1
16. Is there a point in space where an astronaut has no weight? Why or Why not?
17. Why does an astronaut in orbit feel weightless?
18. The mass of Earth’s Moon is M = 7.35 x 1022 kg and its radius is r = 1738 km. Calculate
the acceleration due to gravity at the surface of the Moon.
19. In KSP, the planet Kerbin’s nearest moon is called the Mun. The mass of Kerbin’s
moon, the Mun, is M = 9.76 x 1020 kg and its radius is r = 200 km. Calculate the
acceleration due to gravity at the surface of the Mun.
20. Calculate your own weight on Earth’s Moon, in pounds. Derive a formula to convert
any weight in pounds (on the surface of Earth) to the corresponding weight, in pounds,
on Earth’s Moon. Would your weight be the same on Kerbin’s moon as it is on Earth’s
Moon?
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The Kerbal Math & Physics Lab / Chapter 1
You can check your calculations of your own weight in Newtons, your mass in kilograms, or
your weight on the Moon and on other planets using the following online calculators:
•
•
•
https://www.thecalculatorsite.com/conversions/massandweight.php
https://www.exploratorium.edu/ronh/weight/
http://www.learningaboutelectronics.com/Articles/Weight-on-the-moon-conversioncalculator.php
Thrust-to-Weight Ratio
Suppose the astronaut Jebidiah Kerman weighs 80lbs on the surface of Kerbin. We can
calculate Jeb’s mass in kilograms as about
80 𝑙𝑏𝑠
1
1 𝑘𝑔
× 2.2 𝑙𝑏𝑠 = 36.4 kilograms. On the surface of
Kerbin, with a gravitational acceleration of about 9.8 m/s 2, this means Jeb has a weight of
approximately F = ma = (36.4 kg)(9.8 m/s2) = 357 Newtons. This weight is the force due to
gravity and is directed downwards toward the center of Kerbin. If Jeb is standing on the surface
of Kerbin, an equal and opposite force is provided by the surface of Kerbin, keeping Jeb in
place.
Jeb could jump off the ground if his legs provide greater than 357 Newtons of force, but
without a jetpack or any other propulsive force to provide continued thrust, gravity will bring
Jeb back to the ground.
It is Newton’s third law of motion that expresses the principle that a rocket engine expelling
exhaust in one direction creates a force we call thrust in the opposite direction.
Jeb can achieve and maintain flight with the thrust provided by an airplane or rocket engine.
We define the thrust-to-weight ratio (TWR) as a way to measure when an engine has sufficient
thrust to counteract its own weight, as follows:
𝑇𝑊𝑅 =
𝑡ℎ𝑟𝑢𝑠𝑡
𝑤𝑒𝑖𝑔ℎ𝑡
In this definition both thrust and weight are forces and should be measured in the same units
(for example, Newtons or pounds) and will therefore cancel, meaning that TWR is a
dimensionless value.
A rocket must have a TWR greater than 1 to lift off from the surface of a planet or moon.
As an example, the entire Saturn V rocket that launched the Apollo missions to the moon
weighed about 6.5 million pounds at launch and its five first stage F-1 engines combined to
provide a thrust of about 7.9 million pounds. Therefore, for a Saturn V on the launch pad, we
have
TWR =
𝑡ℎ𝑟𝑢𝑠𝑡
𝑤𝑒𝑖𝑔ℎ𝑡
=
7.9×106 𝑙𝑏𝑠
6.5×106 𝑙𝑏𝑠
= 1.2
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The Kerbal Math & Physics Lab / Chapter 1
A rocket with a TWR equal to 1 would have enough thrust to support its own weight but not
enough to lift that weight, while a rocket with a TWR less than 1 would not be able launch.
TWR values less than or equal to 1 are important during a rocket’s landing phase. For example,
the Apollo Lunar Module needed to support its own weight as it descended to the surface of
the Moon and the SpaceX Falcon 9 booster rockets use TWR values less than 1 to softly touch
down at their landing sites.
We can compute TWR for rocket engines as a useful comparison between different engines. Be
careful to write both thrust and weight in the same units (either pounds or Newtons) and when
using metric, convert mass in kilograms into weight in Newtons. Note also that 1 KiloNewton =
1000 Newtons, and 1 Mega Newton = 1,000,000 Newtons.
Engine
Thrust
(in a vacuum)
1,740,100 lbs
Weight (lbs)
or Mass (kg)
18,499 lbs
TWR
7740.5 KN
8391 kg
2.28 MN
3177 kg
Merlin 1-D (SpaceX)
825 KN
467 kg
RT-5 Solid Rocket Booster
192 KN
1500 kg
215 KN
1500 kg
1350 KN
5000 kg
7740500 N/(8391*9.8 N) =
94.1
2.28x106 N/ (3177*9.8 N) =
73.2
825000 N / (467*9.8 N) =
180.3
192000 N / (1500*9.8) =
13.1
215000 N / (1500*9.8) =
14.6
1350000 N / (5000*9.8) =
2.76
Rocketdyne F-1
(used on Saturn V)
using pounds as units
Rocketdyne F-1
using metric units
RS-25 (Space Shuttle Main
Engine)
(in KSP)
LV-T45 “Swivel”
(in KSP)
Kerbodyne KE-1
(in KSP)
1740100/18499 =
94.1
Note that thrust-to-weight ratio is not a fixed and constant value. For rockets, weight will
change as fuel is burnt, as stages are spent, and as the rocket moves through a changing gravity
field. Also the amount of thrust an engine provides is usually changing during ascent and
descent phases of a flight (during take-off and landing). Finally, we can calculate TWR for jet
engines as well. Note that jet engines do not generally require TWR values greater than 1 to fly
because the speed of the air over the wings of a jet provide lift.
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The Kerbal Math & Physics Lab / Chapter 1
Use the definition, and appropriate units, to compute thrust-to-weight ratios for each of the
following:
1. In Kerbal Space Program, the Jumping Flea, a solid fuel rocket outfitted with a command
pod and parachute, has a total mass of 2440 kilograms and a thrust of 162.9
KiloNewtons at sea level. Calculate the TWR for the Jumping Flea at launch.
2. The Soviet R-7 rocket that launched the Sputnik satellite had a total mass at liftoff of
about 267 metric tons (267,000 kilograms). The four first stage boosters of the R-7
combined to provide a total of 3.89 MegaNewtons of thrust. Calculate TWR for the
Soviet R-7 at launch.
3. The KerbalX, a rocket which can land a Kerbal on the Mun, has a total mass of 128,700
kilograms on the launch pad. At launch, the KerbalX is equipped with six LV-T45 liquid
fuel engines that each provide 168 KiloNewtons of thrust and a single Rockomax
Mainsail engine that provides 1379 KiloNewtons of thrust. Calculate TWR for the
KerbalX at liftoff.
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The Kerbal Math & Physics Lab / Chapter 1
4. The Saturn V, which launched American astronauts to the Moon, had a total mass of
approximately 2.97 million kilograms. It lifted off the launch pad with the combined
thrust of 5 Rocketdyne F-1 engines that each provided about 7.02 million Newtons of
thrust. Calculate the TWR for the Saturn V at liftoff.
5. NASA’s Space Shuttle weighed about 4,470,000 lbs at lift-off. The Shuttle launched with
two solid rocket boosters, each providing about 2,800,000 lbs of thrust, and a main
engine that provided about 1,180,000 lbs of additional thrust. Calculate the TWR for the
Shuttle at lift-off.
6. In February 2018, a test flight of the SpaceX Falcon Heavy launched a red Tesla Roadster
into an orbit about the Sun which reaches a distance further than Mars. At liftoff, the
Falcon Heavy had a mass of about 1.42 million kilograms and 27 Merlin engines each of
which provided about 845,000 Newtons of thrust. Calculate the Falcon Heavy’s TWR at
liftoff.
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The Kerbal Math & Physics Lab / Chapter 1
7. In KSP, the Mun Lander is included with the stock KerbalX rocket and is capable of
landing on the Mun, the nearest moon to Kerbin. Based on the Mun’s mass and radius,
acceleration due to gravity at the surface of the Mun is approximately a = 1.63 m/s2.
Suppose that the Mun Lander has a mass of about 15 metric tons (15,000 kilograms).
What thrust (in Newtons) is required to launch the Mun Lander from the surface of the
Mun?
8. The TWR of a rocket changes during flight as its mass (and thus weight) and also its
thrust change. Consider the following flight data for a solid fuel rocket, based on the
BACC “Thumper” Solid Fuel Booster with the Mk1 Command Pod, an Inline Stabilizer
and a parachute, in KSP.
• Initial mass: 8.69 metric tons
• Initial thrust: 250 KiloNewtons, at sea-level (ASL)
• Final mass: 2.54 metric tons (at engine cut-off)
• Final thrust: 300 KiloNewtons, in a vacuum (VAC)
At launch, acceleration due to gravity on the rocket is 9.8 meters/sec 2 and at engine cutoff, at an altitude of about 30,000 meters above Kerbin, acceleration due to gravity is
approximately 9 meters/sec2.
Compute the rocket’s TWR at liftoff and right before engine cut-off.
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The Kerbal Math & Physics Lab / Chapter 1
A scenario similar to that given previously in question 8 is explored by Mike Aben on YouTube in
his Let’s Do the Math series, available at https://www.youtube.com/watch?v=VS1XACh4upc
and also at https://sites.google.com/kspmath.
In Kerbal Space Program, create a single stage rocket and record mass, weight, thrust, and
thrust to weight ratio (TWR) at launch and before engine cut-off as done by Mike Aben and in
the previous section of this workbook. Check the TWR values you calculate against what is
displayed in game. What is the max TWR encountered by your rocket? How does this value
relate to g-forces experienced by the rocket?
Can you build a rocket with a TWR just above 1 that can hover above the launch pad, like a
SpaceX Falcon 9, Star Hopper or StarShip? Describe your approach, perhaps you can find an
already built ship at KerbalX.com. Describe the challenges of maintaining a TWR at or just
above 1.
The Acceleration and g-Force Lab, Part 1 – The Thumper Launch in KSP
In this lab exercise we’ll examine the thrust-to-weight ratio, delta-v, acceleration and g-force
experienced during a rocket launch.
To illustrate, let’s consider a rocket built in KSP, the “Thumper Test Rocket” made with the
following components: a MK1 command pod, an Advanced Inline Stabilizer, the TD-12
decoupler, the BACC Thumper solid fuel booster and 4 basic aerodynamic stabilizer fins. For this
example, we’ll reduce the BACC Thumper solid fuel booster to 50% thrust and reduce the
amount of fuel in the booster to 500 units.
With 500 units of fuel, the “Thumper” test rocket has a total mass of 6370 kg on the launch pad
and is expected to burn approximately 3750 kg of fuel in 52 seconds, after which the total mass
of the rocket will be about 2580 kg.
At sea-level with 50% thrust, the BACC Thumper engine produces about 125 kiloNewtons of
thrust. Assuming constant thrust during flight, the initial TWR is about 125000/(9.8*6370) = 2
while the final TWR at engine cut-off is approximately 125000/(9.8*2620) = 4.9.
Finally, the estimated ∆𝑣 (delta-v or change in speed) for the Thumper rocket is about 1525
meters/second. This is the theoretical expected change in velocity provided by the rocket,
between launch and engine cut-off, and does not include the effects of gravity and air
resistance. Launching from the surface of Kerbin, with an atmosphere similar to Earth’s, the
actual change in velocity will be less than the theoretical value of 1525 m/s.
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The Kerbal Math & Physics Lab / Chapter 1
In this lab, we will compute TWR, record speed during launch, maximum altitude reached and
compute the rocket’s acceleration and the g-forces experienced by the pilot. The Thumper
rocket is designed to reach space so we’ll launch vertically and then pitch slightly eastward
during ascent to insure we return with a splashdown near the Kerbal Space Center.
There are several ways to record flight data in KSP. The simplest method would be to hit the
escape-key periodically to pause the game and write down speed and altitude. Another
method, that will also track other in-flight data, is to download and install the free game mod
“MechJeb” (short for Mechanical Jeb). With Mechjeb, players have access to a flight data
recorder that can download data in .csv format to analyze in a spreadsheet like Microsoft Excel
or paste into Desmos.
We will use the rocket’s speed over time to estimate its acceleration and the g-forces
experienced by the rocket and pilot.
Acceleration is change in speed over time, so the units are meters/second per second, or m/s 2.
There are two ways we can estimate acceleration based on the data collected:
1. Estimate acceleration (a) by dividing total change in velocity (∆𝑣) by the length of time
of the engine burn (∆𝑡): That is, 𝑎 =
∆𝑣
∆𝑡
.
2. Estimate acceleration (a) by finding the slope of the line of best fit (a linear regression)
for the velocity data.
Both of these methods of calculation provide estimates for the average acceleration over a
given time interval and both can be referred to as estimates of the average rate of change in
velocity. A primary topic in calculus involves finding instantaneous rates of change, for example
the acceleration at a single instant in time. We can return to this lab in chapter 3 to estimate
the instantaneous acceleration on a rocket.
To compute g-force (GF), consider the upward (positive) force N produced by the rocket and the
downward (negative) weight mg of the rocket, where m is the rocket’s mass, and g is the
acceleration due to gravity at the surface of the planet. If the sum, F, of these forces produces
an acceleration (a), we have F = ma = N – mg. Solving for N, we have N = ma + mg. We define
g-force as the ratio of rocket’s thrust to the weight of the rocket. In other words, g-force is
mathematically equal to TWR. and we have
𝐺𝐹 = 𝑇𝑊𝑅 =
𝑁
𝑚𝑎 + 𝑚𝑔 𝑚𝑎 𝑚𝑔 𝑎
=
=
+
= +1
𝑚𝑔
𝑚𝑔
𝑚𝑔 𝑚𝑔 𝑔
Because the factor m drops out in the equation above, we can refer to a rocket’s thrust-toweight ratio, or the g-force experienced by the rocket and pilot, as the same thing.
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The Kerbal Math & Physics Lab / Chapter 1
𝑎
We now have the relation 𝑇𝑊𝑅 = 𝑔 + 1 which relates thrust-to-weight ratio, acceleration, and
surface gravity. As the rocket picks up speed, the acceleration increases as does the thrust-toweight ratio.
Acceleration and g-Force Lab Data:
Build and launch the Thumper Test Rocket as described above (or a similar rocket), and collect
flight data to complete the table below. Record data for at least 10 different points in time
between launch and engine cut-off.
Rocket Name: _______________
Launch Date: ________ Launch Site: ___________
Initial Mass: __________
Engine burn time (in seconds): _______
Total Mass at engine cut-off: _____________
Engine Thrust: ________________
Initial TWR: _________
Final TWR (at engine cut-off): ______________
Theoretical delta-v: ____________ (use values provided in-game or the ∆𝑣 formula in Ch. 2)
Time
(seconds)
Altitude ASL*
(meters)
Speed
(m/s)
Altitude
(miles)
Speed
(miles/hr)
Mach
speed**
* ASL means Above Sea-Level
** For Mach speed, divide speed (in meters/sec) by 343. Mach 1 (343 m/s) is the speed of sound, Mach
2 is twice the speed of sound, in the atmosphere of Earth and Kerbin.
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The Kerbal Math & Physics Lab / Chapter 1
Complete the previous table, or reproduce the table on a separate sheet of paper, and use a
separate sheet of paper to complete the following:
1. Use Excel or Desmos to graph your collected velocity data (from launch to engine cutoff) with seconds on the horizontal axis and speed on the vertical axis.
2. Use Excel or Desmos to graph the height data (from launch to engine cut-off) with
seconds on the horizontal axis and height (in meters) on the vertical axis.
3. Use Excel or Desmos to graph velocity and height data for the entire flight from launch
to splash down.
∆𝑣
4. Estimate the acceleration on the rocket (from launch to engine cut-off) using 𝑎 = ∆𝑡
where ∆𝑣 is the total change in velocity during the burn and ∆𝑡 is the time interval of
the burn.
5. Estimate the acceleration on the rocket (during engine burn) by finding the slope of the
line of best fit for the velocity data over the interval of the engine burn. Use Excel,
Desmos, the calculator or computer to find the line of best fit.
6. Use your estimates for acceleration computed above to estimate the average G-force
experienced by the rocket and pilot between launch and engine cut-off.
7. What is the g-force experienced by the pilot on the launch pad before the engine burn?
8. What is the g-force experienced by the pilot at lift-off immediately after the rocket
engine fires?
9. What is the maximum g-force experienced by the pilot (immediately before engine cutoff)?
10. What is the maximum g-force a human pilot can typically endure before losing
consciousness?
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The Kerbal Math & Physics Lab / Chapter 1
Orbital Speed
Orbital speed for a circular orbit is given by the formula 𝑣 = √
𝐺𝑀
𝑟
, where G is the universal
gravitational constant 6.67 x 10-11 m3/(kg*s2), M is the mass of the object being orbited (like a
Sun, planet or moon) and r is the radius of the orbit. Throughout, we are assuming the mass of
the orbiting body, m, is much smaller than the mass, M, of the object being orbited.
In the formula above, because of the value we use for G, we measure the mass M in kilograms
and the distance r in meters. Also, r represents the entire radius of the orbit, which is the total
distance from the center of the orbiting body to the center of the body being orbited. This
means we need to remember to add the radius of any planet or moon to the altitude of an
orbiting rocket or satellite to determine the full radius of the orbit.
1. A rocket orbits at an altitude of 100 km above the surface of Kerbin which has a radius
of 600 km and a mass of M = 5.29 x 1022 kg. Find the orbital radius of the rocket and
then compute the orbital speed at 100 km.
You can check your answer above with the Kerbal 1 rocket (available in Sandbox mode in KSP).
Check your rocket’s speed in the Navball display and in Map View.
2. What is the orbital speed at 100 km above the surface of Kerbin in miles/hour? Use the
fact that 1 mile = 1609 meters.
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The Kerbal Math & Physics Lab / Chapter 1
3. What is the orbital speed at 100 km above the surface of the Earth in miles/hour and
also in meters/sec? Use the facts that the mass of the Earth is M = 5.97 x 1024 kg and its
radius is approximately r = 6,378,000 meters.
4. Compute the acceleration due to gravity an altitude of 100 km above Kerbin. What is
the force of gravity (in Newtons) on a ship with a mass of 3000 kg at this altitude?
5. Does orbital speed in a circular orbit depend on the mass of the orbiting body? What is
the orbital speed for a Kerbal astronaut on EVA outside a spacecraft orbiting in a circular
orbit at 100 km above Kerbin?
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The Kerbal Math & Physics Lab / Chapter 1
6. Does orbital speed depend on the size of the planet or moon being orbited? Explain.
7. The Mun is the closest moon to the planet Kerbin and orbits in a circular orbit at an
altitude of 11,400 kilometers above the surface of Kerbin. Determine the orbital radius
of the Mun. Compute the orbital speed of the Mun. This value can be confirmed in
Map View in the game.
8. If a rocket increased its altitude to a circular orbit at 150 km above the surface of Kerbin,
what is the orbital speed at this higher altitude? Is it traveling faster or slower than it
was at 100 km altitude?
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The Kerbal Math & Physics Lab / Chapter 1
9. The mass of the Mun is 9.76 x 1020 kg and its radius is 200 km. Compute the orbital
speed of a rocket in a circular orbit around the Mun at an altitude of 13 km.
Build a rocket in KSP to reach the Mun or use the stock KerbalX rocket available in sandbox
mode. There are many tutorials on reaching the Mun available on YouTube, including some by
Mike Aben at sites.google.com/view/kspmath and detailed instructions on the KSP Wiki, on
how to reach a Mun orbit in KSP.
During flight, check orbital speed in flight above Kerbin and above the Mun to compare with the
previous calculations.
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The Kerbal Math & Physics Lab / Chapter 1
Orbital Period
For a circular orbit, we can determine orbital period using the equation 𝑇 =
2𝜋𝑟
𝑣
where T is the
period (in seconds) of the orbit, r is the radius of the orbit (in meters), and v is the orbital speed
(in meters/sec). Show all work required to answer the following.
1. Find the period of a circular orbit about Kerbin at an altitude of 100km above the
surface. Write the orbital period in terms of minutes and seconds.
2. The International Space Station orbits above the Earth in nearly circular orbit at an
altitude of approximately 250 miles. Compute the orbital speed and period of the ISS.
Write the speed in meters/sec and miles/hr and write the period in minutes and
seconds.
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The Kerbal Math & Physics Lab / Chapter 1
3. The Mun’s orbital radius about the center of Kerbin is exactly 12,000 km and the Mun’s
orbital speed is 542.5 m/s. Compute the period of the Mun’s orbit about Kerbin.
The planet Kerbin orbits around a sun called Kerbol. The mass of Kerbol is M = 1.76 × 1028 kg
and Kerbin orbits at a radius of approximately 1.36 x 1010 meters from its sun.
4. Use the values above to calculate Kerbin’s orbital speed about its sun.
5. Find the period of Kerbin’s orbit about the sun. This value represents the approximate
length of a Kerbal year.
Check answers in game using Map View or at wiki.kerbalspaceprogram.com/wiki/Main_Page.
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The Kerbal Math & Physics Lab / Chapter 1
The previous discussion dealt only with circular orbits. We now consider the period of an
elliptical orbit. Kepler’s third law of planetary motion states the square of the orbital period for
any elliptical orbit is proportional to the cube of the semi-major axis of the orbit.
Mathematically, this can be written as
2
𝑇 =
4𝜋2
𝐺𝑀
𝑎3
Here T is the orbital period, a is half of the length of the larger axis of the ellipse as shown in the
diagram above, G is the universal gravitational constant, and M is the mass of the primary body
being orbited. In this form of the equation we assume the mass of the body being orbited is
significantly larger than the orbiting body, like a planet that orbits about a sun, or a rocket
orbiting a planet or moon. Also, if we use G = 6.67 x 10-11 m3/(kg*s2) then the units of T are
seconds, a is measured in meters and M is measured in kilograms.
6. In Kepler’s third law as written above, what is the constant of proportionality?
7. Solve for T in the equation describing Kepler’s third law and write a formula for orbital
period in an elliptical orbit.
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The Kerbal Math & Physics Lab / Chapter 1
8. In the Kerbal system, the planet Duna orbits in an elliptical orbit with a semi-major axis
of a = 2.07 x 1010 meters around its sun Kerbol which has a mass of 1.758 x 1028 kg. Use
Kepler’s third law to find the orbital period of the planet Duna.
9. Suppose that astronomers on the planet Kerbin discover the planet Eeloo orbiting about
the sun with a period of approximately 1.57 x 108 seconds. Use Kepler’s third law to
determine the semi-major axis of Eeloo’s orbit.
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The Kerbal Math & Physics Lab / Chapter 1
Satellites in geostationary orbit remain in a fixed position above the surface of the Earth. This
has useful applications for communications, weather observation and navigation. To remain in
a fixed position relative to a point on Earth, as the Earth rotates on its own axis, the period of a
geostationary orbit must be approximately 23 hours, 56 minutes and 4 seconds. This is the
time it takes the Earth to rotate about its axis, relative to the apparent fixed background of the
stars, and is defined as the length of a sidereal day.
Geosynchronous orbits, like geostationary orbits, also have an orbital period equal to the length
of a sidereal day. While satellites in geostationary orbit remain at a fixed point relative to the
surface of the Earth and require a circular orbit directly above the equator, geosynchronous
orbits can be elliptical or inclined relative to the equator.
10. Use the equation 𝑇 2 =
4𝜋2
𝐺𝑀
𝑎3 to solve for a, the semi-major axis of an elliptical orbit, in
terms of the period, T, and mass, M, of the primary body.
11. Determine the number of seconds in the orbital period of a geosynchronous orbit. Use
this result above to determine the semi-major axis of a geosynchronous orbit.
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The Kerbal Math & Physics Lab / Chapter 1
12. A geostationary orbit (GEO) is a circular orbit with a radius equal to the semi-major axis
of a geosynchronous orbit. Calculate orbital speed for a geostationary orbit.
13. Look up the length of a sidereal day on Kerbin. Use the result to find the radius of a
Kerbin-stationary (KEO) orbit.
14. What is orbital speed for a satellite in Kerbin-stationary orbit?
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The Kerbal Math & Physics Lab / Chapter 1
Computing the Mass of a Planet or Moon
Assuming an object is in a circular orbit, we know orbital speed is given by 𝑣 = √
𝐺𝑀
𝑟
, where v
is the speed in meters/sec, M is the mass of the body being orbited, r is the orbital radius and G
is the universal gravitational constant 6.67 x 10-11.
1. Use the formula for orbital speed in a circular orbit to solve for M.
2. The Mun orbits Kerbin at 542.5 m/s in a circular orbit with an orbital radius of 12,000
km. Use these values to compute the mass of Kerbin.
3. Kerbin orbits its sun, Kerbol, at a speed of 9285 m/s in a circular orbit with a radius of
13,599,840,256 meters. Use this to compute the mass of the Sun in the Kerbal system.
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The Kerbal Math & Physics Lab / Chapter 1
4. Given the period of a circular orbit can be expressed as 𝑇 =
2𝜋𝑟
𝑣
where r is the orbital
radius and v is the linear speed, find an expression for the linear speed in a circular orbit
in terms of orbital radius and period.
5. Given that Moon orbits about Earth at an orbital radius of about 239,000 miles with an
orbital period of approximately 27.3 days, estimate the linear speed of the Moon’s orbit
about Earth and use this result to estimate the mass of the Earth.
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The Kerbal Math & Physics Lab / Chapter 1
The vis-viva equation relates the speed of an object in an elliptical orbit, with distance from the
primary body, the length of the semi-major axis, and the mass of the body being orbited. One
form of this equation is given below
2 1
𝑣 2 = 𝐺𝑀 ( − )
𝑟 𝑎
In this equation, v is speed (in meters/sec), G is the universal gravitational constant, M is the
mass of the body being orbited (in kg), r is the distance (in meters) between the centers of mass
of the two bodies and a is the semi-major axis of the ellipse (also in meters).
6. Use the above equation to solve for M and find a formula for the mass of a primary
body based on speed, distance and semi-major axis of another object in orbit about the
primary body.
7. A KerbalX rocket is in an orbit above the Mun at an altitude of 70 kilometers at a speed
of about 492.2 meters/sec. Suppose the semi-major axis of the orbit is a = 271,111
meters. Recall that the gravitational constant is G = 6.67 x 10-11 and the Mun’s radius is
200 km. Compute the mass of the Mun.
Confirm the value for the mass of the Mun either by looking up the value on the Kerbal
Wiki or checking the map view in the game.
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The Kerbal Math & Physics Lab / Chapter 1
Kinetic and Potential Energy in a Circular Orbit
1
Kinetic energy (KE) is given by the formula 𝐾𝐸 = 2 𝑚𝑣 2 and potential energy (PE) is given by
the formula 𝑃𝐸 =
−𝐺𝑀𝑚
𝑟
where G is the universal gravitational constant, r is the distance
between the centers of mass of two bodies, M the larger mass being orbited (the sun, a planet
or moon) and m, the smaller mass of the orbiting body (a planet, rocket, satellite or moon).
Recall that velocity in a circular orbit is given by the equation 𝑣 = √
𝐺𝑀
𝑟
.
1. Use algebraic substitution to show that, for a circular orbit, the kinetic energy of an
orbiting body is exactly half the absolute value of its potential energy.
2. The Mun orbits Kerbin at an orbital radius of r = 1.2 x 107 meters. The Mun’s orbital
speed is 543 m/s. The Mun’s mass of 9.76 x 1020 kg and Kerbin’s mass is 5.29 x 1022 kg.
Compute kinetic and potential energy for the Mun. How do these values relate?
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The Kerbal Math & Physics Lab / Chapter 1
3. Launch a rocket into a circular orbit about Kerbin. Use the MechJeb mod or a
spreadsheet to determine kinetic and potential energy at different radius orbits.
Complete the table below and describe the relationship between kinetic and potential
energy in a circular orbit.
Orbit
orbital
radius (r)
rocket
mass (m)
kinetic energy (KE)
potential energy (PE)
100 km
altitude
1000 km
altitude
2863.33 km
altitude
(KEO)
34