Physics with KSP 1

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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 
, 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 
 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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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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As we have seen, acceleration due to gravity can be computed with the equation   
 .
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
.
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,
  m/s2
and for Earth, with a mass M = 5.97 x 1024 kg and radius r = 6,378,000 meters,
 m/s2

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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
 Newtons
For Earth, at an orbital radius of r = 6,378,000 + 100,000 = 6,478,000 meters
 Newtons
Use the previously introduced unit conversions and the equation   
 to complete the
following:
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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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.