“This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent powerful Being” – Sir. Issac Newton
When I was seven years old, I had the notion that all planets dance around the sun along a wavy orbit (see Fig. 1).
Many years later, I took on a challenge to show mathematically the orbit of my ‘dancing planet’ . This post is a long overdue report of my journey.
Shown in Fig. 2 is the sun and a planet in a x-y-z coordinate system. The sun is at the origin. The moving planet’s position is being described by .
According to Newton’s theory, the gravitational force sun exerts on the planet is
where is the gravitational constant, the mass of the sun and planet respectively. .
By Newton’s second law of motion,
(0-3) (0-2) yields
it must be true that
where is a constant.
where are constants.
If then by the following well known theorem in Analytic Geometry:
“If A, B, C and D are constants and A, B, and C are not all zero, then the graph of the equation Ax+By+Cz+D=0 is a plane“,
(0-7) represents a plane in the x-y-z coordinate system.
For , we have
where is a constant. Simply put,
Hence, (0-7) still represents a plane in the x-y-z coordinate system (see Fig. 3(a)).
The implication is that the planet moves around the sun on a plane (see Fig. 4).
By rotating the axes so that the orbit of the planet is on the x-y plane where (see Fig. 3), we simplify the equations (0-1)-(0-3) to
It follows that
Integrate with respect to ,
where is a constant.
We can also re-write (0-6) as
where is a constant.
Using polar coordinates
we obtain from (1-2) and (1-3) (see Fig. 5):
If the speed of planet at time is then from Fig. 6,
Suppose at , the planet is at the greatest distance from the sun with and speed . Then the fact that attains maximum at implies . Therefore, by (1-4) and (1-5),
We can now express (1-4) and (1-5) as:
By chain rule,
Notice that .
By doing so, (1-14) can be expressed as
Take the first case,
Integrate it with respect to gives
where is a constant.
By (1-11), it is
Solving (1-15) for yields
Studies in Analytic Geometry show that for an orbit expressed by (1-16), there are four cases to consider depend on the value of :
We can rule out parabolic and hyperbolic orbit immediately for they are not periodic. Given the fact that a circle is a special case of an ellipse, it is fair to say:
The orbit of a planet is an ellipse with the Sun at one of the two foci.
In fact, this is what Kepler stated as his first law of planetary motion.
from which we obtain
This is an ellipse. Namely, the result of rotating (1-16) by hundred eighty degrees or assuming attains its minimum at .
The second case
can be written as
Integrate it with respect to yields
from which we can obtain (1-16) and (1-17) again.
Over the time duration , the area a line joining the sun and a planet sweeps an area (see Fig. 9).
is a constant. Therefore,
A line joining the Sun and a planet sweeps out equal areas during equal intervals of time.
This is Kepler’s second law. It suggests that the speed of the planet increases as it nears the sun and decreases as it recedes from the sun (see Fig. 10).
Furthermore, over the interval , the period of the planet’s revolution around the sun, the line joining the sun and the planet sweeps the enire interior of the planet’s elliptical orbit with semi-major axis and semi-minor axis . Since the area enlosed by such orbit is (see “Evaluate a Definite Integral without FTC“), setting in (2-1) to gives
The stages of a two stage rocket have initial masses and respectively and carry a payload of mass . Both stages have equal structure factors and equal relative exhaust speeds. If the rocket mass, , is fixed, show that the condition for maximal final speed is
Find the optimal ratio when .
According to multi-stage rocket’s flight equation (see “Viva Rocketry! Part 2“), the final speed of a two stage rocket is
Let , we have
Differentiate with respect to gives
It follows that implies
That is, . i.e.,
It is the condition for an extreme value of . Specifically, the condition to attain a maximum (see Exercise-2)
When , solving
yields two pairs:
Only (3) is valid (see Exercise-1)
The entire process is captured in Fig. 2.
Exercise-1 Given , prove:
Exercise-2 From (1), prove the extreme value attained under (2) is a maximum.
The stages of a two-stage rocket have initial masses and respectively and carry a payload of mass . Both stages have equal structure factors and equal relative exhaust speed . The rocket mass, is fixed and .
According to multi-stage rocket’s flight equation (see “Viva Rocketry! Part 2“), the final speed of a two-stage rocket is
Let , it becomes
where . We will maximize with an appropriate choice of .
That is, given
where . Maximize with an appropriate value of .
The above optimization problem is solved using calculus (see “Viva Rocketry! Part 2“). However, there is an alternative that requires only high school mathematics with the help of a Computer Algebra System (CAS). This non-calculus approach places more emphasis on problem solving through mathematical thinking, as all symbolic calculations are carried out by the CAS (e.g., see Fig. 2). It also makes a range of interesting problems readily tackled with minimum mathematical prerequisites.
The fact that
is a monotonic increasing function
(1) can be written as
for gives if .
Hence, (1) is a quadratic equation. For it to have solution, its discriminant must be nonnegative, i.e.,
If , (3) is a quadratic equation.
Solving (3) yields two solutions
and, the solution to (2) is
We prove that (4) is true by showing (5) is false:
It can be written as
Since (see Exercise 1) and,
solve (7) for yields
It follows that for .
Consequently, is a negative quantity. i.e.,
which tells that (5) is false.
Hence, when , the global maximum is .
Solving for :
attains maximum at .
In fact, attains maxima at even when , as shown below:
Solving for , we have
Only is valid (see Exercise-2),
Solve quadratic equation for yields
The coefficient of in is , a negative quantity (see Exercise-3).
The implication is that is a negative quantity when .
A rocket with stages is a composition of single stage rocket (see Fig. 1) Each stage has its own casing, instruments and fuel. The th stage houses the payload.
The model is illustrated in Fig. 2, the stage having initial total mass and containing fuel . The exhaust speed of the stage is .
The flight of multi-stage rocket starts with the stage fires its engine and the rocket is lifted. When all the fuel in the stage has been burnt, the stage’s casing and instruments are detached. The remaining stages of the rocket continue the flight with stage’s engine ignited.
Generally, the rocket starts its stage of flight with final velocity achieved at the end of previous stage of flight. The entire rocket is propelled by the fuel in the casing of the rocket. When all the fuel for this stage has been burnt, the casing is separated from the rest of the stages. The flight of the rocket is completed if . Otherwise, it enters the next stage of flight.
A rocket is programmed to burn and ejects its propellant at the variable rate , where and are positive constants. The rocket is launched vertically from rest. Neglecting all external forces except gravity, show that the final speed given to the payload, of mass , when all the fuel has been burnt is
Here is the speed of the propellant relative to the rocket, the initial rocket mass, excluding the payload. The initial fuel mass is .
Exercise 2. Before firing, a single stage rocket has total mass , which comprises the casing, instruments etc, with mass , and the fuel. The fuel is programmed to burn and to be ejected at a variable rate such that the total mass of the rocket at any time , during which the fuel is being burnt, is given by
where is a constant.
The rocket is launched vertically from rest. Neglect all external forces except gravity, show that the height attained at the instant the fuel is fully consumed is
It follows that the percentage reduction in the predicted final speed due to the inclusion of gravity is
Using the given values which are typical, the estimated value of (3) (see Fig. 2) is
This shows the results obtained without taking gravity into consideration can be regarded as a reasonable approximation and the characteristics of rocket flight indicated in “Viva Rocketry! Part 1” are valid.
Since , (1) can be written as
To find the distance travelled while the fuel is burnt, we solve yet another initial-value problem:
The solution (see Fig. 3) is
Hence, the height reached at the burnt out time is
Using the given values, we estimate that (see Fig. 4)
Exercise 1: Find the distance the rocket travelled while the fuel is burnt by solving the following initial-value problem:
Shown in Fig. 1 is an experimental car propelled by a rocket motor. The drag force (air resistance) is given by . The initial mass of the car, which includes fuel of mass , is . The rocket motor is burning fuel at the rate of with an exhaust velocity of relative to the car. The car is at rest at . Show that the velocity of the car is given by, for ,
where , and is the time when the fuel is burnt out.