Force, Motion and Work - Grades 11/12

• use vectors to represent force, velocity, and acceleration
• analyze quantitatively the horizontal and vertical motion of a projectile
• identify the frame of reference for a given motion
• apply Newton's laws of motion to explain inertia, the relationship between force, mass, and acceleration, and the interaction of forces between two objects
• analyze quantitatively the relationship among force, distance, and work
• analyze quantitatively the relationship among work, time, and power
• analyze quantitatively two-dimensional motion in a horizontal plane and a vertical plane
• describe uniform circular motion, using algebraic and vector analysis
• explain quantitatively circular motion using Newton's laws
• select and use appropriate numeric, symbolic, graphical, and linguistic modes of representation to communicate ideas, plans, and results

Energy and Momentum - Grades 11/12

• analyse quantitatively the relationships among mass, height, speed, and heat energy using the law of conservation of energy
• apply quantitatively Newton's laws of motion to impulse and momentum
• provide a statement that addresses the problem or answers the question investigated in light of the link between data and the conclusion
• select and use appropriate numeric, symbolic, graphical, and linguistic modes of representation to communicate ideas, plans, and results
• identify multiple perspectives that influence a science-related decision or issue

Radioactivity and Modern Physics - Grades 11/12

• explain quantitatively the Bohr atomic model as a synthesis of classical and quantum concepts
• explain the relationship between the energy level in Bohr's model, the energy difference between the levels, and the energy of the emitted photons
• describe the products of radioactive decay and the characteristics of alpha, beta, and gamma radiation
• describe sources of radioactivity in the natural and constructed environments
• compare and contrast qualitatively and quantitatively nuclear fission and fusion

Science, Technology, Society and the Environment - Grades 11/12

• analyse and describe examples where technologies were developed based on scientific understanding
• describe and evaluate the design of technological solutions and the way they function, using scientific principles
• identify new questions or problems that arise from what was learned
• identify various constraints that result in tradeoffs during the development and improvement of technologies

Of course the shape may very well have to comply with more utilitarian requirements such as strength, simplicity, reliability and cost.

The two rockets shown below are typical shapes that we associate with rockets launched from the Earth's surface.

Rocket engines all operate on the same principle; namely,

the law of conservation of momentum (P).

We will use the symbol P to represent momentum, and define momentum as the vector product of mass times velocity. i.e. P = mv.

A change in an object's momentum (DP) can be computed from the physical quantity called Impulse or from the object's change in velocity DV (provided that its mass is constant).

We define impulse I as,

I = FDt

where F is the thrust (force) in newtons, and Dt is the duration that the force acts (in seconds).

The following equivalencies are useful,

impulse = change in momentum

Transparency Master

The propellant and rocket-body/payload make up a system whose momentum will remain zero if external forces do not intervene.

If the propellant is ejected in one direction with a momentum P then the rocket/payload will gain momentum -P, (the negative sign meaning that the momentum vector is directed in the opposite direction to that of the exhaust gases) so that the momentum of the system remains zero.

Recall that momentum is defined as the product of mass times velocity.

P=mv

IF all of the propellant were to be ejected instantaneously, the momentum gained by a spacecraft would be equal to the mass of the propellant times the ejected velocity of the propellant. Sadly, it's not quite that simple.

The reason that it is not simple is because all the propellant is NOT ejected instantaneously, but it is ejected gradually as the fuel burns. This means that some of the momentum of the exhaust gases (P) is actually being transferred (-P) to the unused fuel, and not just to the spacecraft.

The chart shown gives the D v's required for both Lunar and Mars missions.

The challenge for aerospace engineers is to achieve the required D v's using the minimum amount of energy ... since energy equates to cost in a very unfavourable way.

The further and faster the spacecraft must travel, the more propellant it must carry. This places very large constraints on the kinds of practical interplanetary missions that can be carried out.

The best rocket engine is the one that makes the most efficient use of the propellant; that is, the one which transfers the most momentum to the spacecraft, per unit of propellant used.

A typical liquid-hydrogen liquid-oxygen rocket, designed to place a satellite in Low Earth Orbit (LEO) has a mass ratio of about 7.6 to 1, meaning that the mass of the propellant is 7.6 times greater than the mass of the payload (satellite or crew module).

1. optimizing the shape of the rocket's exhaust nozzle.
2. choosing a propellant with low molecular mass so that the exhaust velocity at a given temperature is maximized.
3. increasing the exhaust gas temperature.

Each of these methods has limitations.

In the late 1800's and early 1900's the Russian scientist and visionary Konstantin Eduardovich Tsiolkovsky formulated what has come to be known as the "Rocket Equation".

The rocket equation relates the mass of propellant needed m_p to give a spacecraft of mass m_c a change in velocity DV based on the rocket engine's exhaust velocity V_e

Transparency Master

The key to building a really successful rocket is to select a propellant that imparts the maximum exhaust velocity to the gases being ejected through a suitably designed rocket nozzle.

Various propellants and rocket engine designs have been developed in an attempt to reduce the required propellant load as much as possible while maximizing the allowed payload mass.

Details of rocket design can be found in the document Rocket Theory.

One of the best is the liquid-hydrogen (fuel) and liquid-oxygen (oxidizer) combination.

An analysis of the rocket equation however shows that for very large DVs the mass ratio becomes unacceptably large even for rockets using the liquid-hydrogen and liquid-oxygen combination.

m_p/m_c ratios (from the Rocket Equation)

Transparency Master

Solid propellant rockets suffer from the high molecular mass of their exhaust gases, which means lower exhaust velocities. Solid propellants are however much easier to handle than liquid cryogenically cooled propellants.

If raw hydrogen atoms could be heated to very high temperatures, without having to react them with oxygen, then they would make an excellent propellent and the mass ratio for a rocket could be substantially reduced. To achieve this nuclear powered rockets have shown considerable promise.

Although a nuclear rocket could also be called "A HOT HYDROGEN ROCKET".

Since a nuclear reactor can heat hydrogen to a very high temperature, a nuclear rocket has the potential to produce extremely high exhaust velocities; therefore, the mass-to-propellant ratio for nuclear rockets would be much less than it is for conventionally powered rocket engines.

A reduction in the mass-to-propellant ratio means that the mass of the payload can be increased.

The design of a nuclear rocket engine is basically quite simple. The reactor core is composed of multiple fuel rods containing plutonium or enriched uranium. The spontaneous decay of atoms in the fuel rods produces a weak background of fast neutrons.

Of course there is a slight mass penalty for nuclear rockets when humans are concerned. Astronauts must be protected with suitable radiation shielding. However this mass penalty is more than compensated for by the huge increase in efficiency due to the improved mass ratio.

The source of energy for a nuclear rocket comes from the fission of heavy nuclei.

The physics which describe this process is given by Einstein's famous equation,

It is the heat produced that can be used to power a rocket.

This process of natural radioactive decay causes elements to gradually undergo a series of transformations ultimately leading to lighter more stable elements.

As well as the production of fast neutrons, fission also releases a huge amount of thermal energy.

The slow neutrons interact with the atoms in the fuel rods and initiate a chain reaction that increases rapidly. Uncontrolled the reaction could accelerate so quickly that a nuclear explosion could ensue.

The reaction can be "throttled" by extracting the moderator rods and also by inserting control rods containing cadmium atoms which readily absorb neutrons.

While passing through the reactor core the hydrogen gas becomes superheated and escapes through the nozzle at the opposite end of the reactor.

The escaping gas can be heated to temperatures far in excess of the temperatures produced by chemical means. The high gas temperatures provide huge gains in exhaust velocity and a corresponding gain in efficiency.

Student Activity 1: Test Your Understanding

Momentum & Impulse

1. A small model rocket has a mass of 120g (0.120kg). What is its momentum if its speed is 22m/s?

2. A small rocket has a fixed mass m and it is acted upon by a force for a time interval . At the end of the time interval its change in velocity is . (Assume all motion lies in a straight line). Consider the following cases.
   1. ) If everything stays the same, except that the time interval is doubled, then the change in velocity is
      1. unchanged, i.e. .
      2. doubled (two fold), i.e. 2
      3. tripled (three fold), i.e. 3
      4. quadrupled (four-fold), i.e. 4
      5. one half, i.e. /2

   2. ) If everything stays the same, except that the mass of the rocket is doubled, then the change in speed will be
      1. unchanged, i.e. .
      2. doubled (two fold), i.e. 2
      3. tripled (three fold), i.e. 3
      4. quadrupled (four-fold), i.e. 4
      5. one half, i.e. /2

   3. ) If everything stays the same, except that the force is doubled, then the change in velocity will be
      1. unchanged, i.e. .
      2. doubled (two fold), i.e. 2
      3. tripled (three fold), i.e. 3
      4. quadrupled (four-fold), i.e. 4
      5. one half, i.e. /2

   4. ) If everything stays the same, but the rocket is taken to Mars where its weight is only one third that of its Earth weight, then its change in velocity is
      1. unchanged, i.e. .
      2. doubled (two fold), i.e. 2
      3. tripled (three fold), i.e. 3
      4. quadrupled (four-fold), i.e. 4
      5. one third, i.e. /3.

3. Fuel and oxidizer, whose total mass is 875 grams (0.875kg) is ejected downward from a small rocket with a net downward momentum of 3.3kg· m/s. What is the upward velocity attained by the rocket if its mass is 150 grams (0.150kg)?

4. The combustion products of a rocket engine have a total mass of 900 grams (0.900kg) and are ejected downward from a rocket nozzle with a net downward impulse of 5.0Ns. What is the upward velocity attained by the rocket if its mass is 150 grams (0.150kg)?

5. The combustion products of a rocket engine having a total mass of 1300 grams (1.300kg) is ejected downward from a small rocket with a net downward momentum of 12.0 kg· m/s. What is the upward velocity attained by the rocket if its mass is 220 grams (0.150kg)?

6. Show, by dimensional analysis, that Momentum ( M=mv) has the same units as Impulse ( I=Ft).

7. A 150 gram rocket experiences an average force of 330N for 35ms. What speed is achieved by the rocket?

8. A 120 gram rocket experiences an average force of 330N for 15ms. What speed is achieved by the rocket?

9. A 240 gram rocket experiences an average force of 330N for 15ms. What speed is achieved by the rocket?

10. A 150 gram rocket experiences an average force of 165N for 35ms. What speed is achieved by the rocket?

Answer Key

Traveling from the Earth to Mars and back is a long arduous journey which takes many months, in fact, several years are needed to make the round trip.

Various flight plans have been developed to accomplish the trip with the least expense in the least amount of time.

One such plan uses a "gravity assist" maneuver (during an encounter with Venus) to hurl the spacecraft speedily onward to Mars. The plan is unusual in that it first deceases the spacecraft velocity so that it falls inwards towards the Sun, rather than increasing its velocity, which is the requirement for a more direct mission to Mars.

Even with the best technology, the only practical flight paths for interplanetary travel are ballistic, meaning that a spacecraft is given a short "boost" to change its speed and then it simply coasts to its destination, much like throwing a baseball.

Mission Summary

No two trajectories are alike!

In fact years pass between the times when a launch window opens that provides a flight that is both economical and reasonably short, say less than three years.

A Venus fly-by mission to Mars requires that the planets (in this case the Earth, Venus, and Mars) be correctly aligned so that the flight path of the spacecraft takes into account the motion of the planets during the voyage.

The intensity of solar cosmic rays (mostly protons) is also greatly increased as one approaches the orbit of Venus (i.e. closer to the Sun).

The diagrams which follow illustrate the flight path of a human mission to the planet Mars, based on the TV mini series Race to Mars.

1. Circles represent the position of the planets
   • yellow (centre) the Sun
   • light blue (inner most orbit) Venus
   • light green (Earth)
   • orange (outermost) Mars.
2. The red arrow represents the spacecraft trajectory.
3. Crosses, (+) represent the position of each planet at the moment the spacecraft leaves Low Earth Orbit (LEO) and begins its interplanetary journey to Mars.
4. X's, (X) represent the position of each planet at the moment the spacecraft leaves Low Mars Orbit (LMO) and begins the return journey to Earth.

Launch to Mars

The launch direction is "backwards" with respect to the Earth's orbital motion.

Day 60

This causes it to "fall" inwards towards the Sun.

At first the spacecraft which is slowed down begins to fall behind (the Earth) as the Earth speeds away in its orbit.

Day 120

After a few weeks the trans-Martian spacecraft has caught up with the Earth and begins to overtake it.

Rapidly gaining speed it plunges onwards towards Venus.

Rendezvous with Venus

By carefully adjusting the flight path, the spacecraft becomes (temporarily) gravitationally attached to the planet Venus.

Attracted by Venus, the spacecraft gains an enormous velocity. The spacecraft has "stolen" kinetic energy from the orbital energy of Venus.

Day 180

Having encountered Venus the spacecraft has changed both speed and direction.

With its huge gain in velocity, the spacecraft shoots past Venus, and slips into a new interplanetary orbit, headed for Mars.

Day 240

If the velocity from Venus has been correctly chosen, the spacecraft and Mars will be convergent trajectories.

Rendezvous with Mars

It is going much faster than Mars and gravitational capture is not possible unless retro-rockets and aerobraking are used to dissipate the kinetic energy of the spacecraft.

The Search for Water

Mars is also intensely cold.

Surviving on Mars would be made much easier if a source of water could be located and utilized by those intending to stay on the surface of the planet.

Locating water is an essential element in the pursuit of a long-term Martian exploration.

Whoever finds (a suitable source of) water first will virtually own Mars.

Day 352

As in the case of leaving the Earth for an encounter with Venus, leaving Mars for an encounter with the Earth means launching "backwards" from the orbital motion of Mars.

This causes the spacecraft to lose energy. It will then begin to "fall" sunward towards the orbit of the Earth.

Day 412

The Sun's gravity pulls the spacecraft towards the Earth's orbit.

As it nears the Earth's orbit the spacecraft accelerates.

Day 472

The Earth, in its orbit is moving considerably faster than the spacecraft and eventually overtakes and passes the spacecraft.

Nevertheless, the spacecraft is constantly accelerating under the gravitational tug of the Sun.

Day 532

Rendezvous with Earth

However the spacecraft is going too fast to be gravitationally captured by the Earth, therefore, as in the rendezvous with Mars, gravitational capture is not possible unless retro-rockets and aerobraking are used to dissipate the kinetic energy of the spacecraft.

1. To fully appreciate the nature of flight planning and trajectory design, print out the transparency masters for each of the journey intervals (called trajectory maps) shown above. Then, using a coloured high-lighter draw in the distance that each planet has moved in the time interval indicated on the page.

   Arrange the pages on a bulletin board in the correct serial order.

   1. Divide the class into 12 teams (one for each trajectory map beginning at Day 60). Assign one trajectory map to each team beginning at Day 60.

      Have each team design a simple daily menu for a crew of six. Estimate the mass (in kilograms) of all the food products consumed (in one day) including water.

      Assuming that each day's food-mass requirements are roughly the same, regardless of the details of the specific menu, calculate the total mass (in kilograms) of all food products consumed during the time interval from the previous trajectory map to their trajectory map.

   2. Add up the total food-mass requirements for all twelve teams.

      Discuss methods whereby this mass could be reduced.

2. During the Venus slingshot fly-by maneuver the spacecraft undergoes an enormous acceleration. What does the crew experience during this maneuver?

3. During the Venus fly-by the spacecraft increases its velocity enormously and as a result there is a huge gain in its kinetic energy. Where does this energy come from?

4. In the context of this mission what would be the consequences of staying on Mars for more than 60 days?