1.    The first point to observe is that kinetic energy is calculated using the formula:

K.E. = 1/2mv²

Here you don't have the velocity given to you. Instead you are given the distance and time. Assuming that the Nimitz is travelling at constant velocity, you can write

Velocity v = distance d/time t

Now the next point you have to observe is that velocity must be in m/s if you want K.E. in Joules.

So you can now write

v= 208000/3.5x60x60

You've converted 208 km = 208000 m and 3.5 h= 3.5x60 = 210 min = 210x60 = 12600 secs

Therefore the velocity of the Nimitz must be v = 208000/12600 = 16.5 m/s approximately.

Now you can calculate the kinetic energy as follows:

K.E. = 1/2 x (8.35 x 10⁷) x (16.5)² = 1.14 x 10¹⁰ Joules.

Note: The problem has three significant figures and you too have given your answer to three significant figures. This is very important to understand. Just because you have an accurate calculator, you cannot give your answer to several significant figures.

I always encourage my students to finally say give their answer written down at the end of the problem and say for example here

K.E. of the Nimitz = 1.14 x 10¹⁰ J (3 s.f.)

2.    I have answered in detail another question on this issue and would recommend you go through his questions and my answers on this issue for more clarification.

Meanwhile, understand that friction is not as simple as it sounds. For practical purposes it is true that friction is proportional to the NORMAL REACTION and my answers to Yang have gone into detail how.

If you push vertically you are not having any component parallel to the object to try and move it along the rough plane. Physically the work done is zero because there is NO DISPLACEMENT in the direction of the force.

If the object is already moving as you suggest in your question, then friction will definitely DEPEND on the force you applied PLUS the weight.

You are however spending some of your chemical energy in trying to push the object and are likely to get tired after a while.

Friction as I said earlier is not simply a mere matter of interlocking of the molecules obstructing motion. If that were the case then you would expect ZERO FRICTION between two surfaces which are perfectly smooth. In fact experiments done in space with little dust on smooth surfaces show an INCREASE in friction! Obviously electromagnetic forces between the molecules are coming into play here but for our practical purposes for most applications FRICTION IS PROPORTIONAL TO THE NORMAL REACTION.

3.    There are basically three concepts here. The first is using the equation of motion

v² = u² + 2a.d

and the other is using Newton's second law

F = m.a

and the third is to use the formula for for work which is

Work = F.d

Let us assume that the object starts from rest. Then the initial velocity would be

u = 0 and the above equation becomes

    v² = 2a.d

From here if you make a the subject of the equation

    a = (v²/2d)

Using NEWTON'S SECOND LAW

    F = mv²/2d

Subsituting this in the equation for work you'll have

W = F.d
    = [mv²/2d].d

    = 1/2 mv²

This work done on the object is stored as kinetic energy.

4.    The pulley as you know is one of six kinds of a simple machines, a lever, gear, screw, wheel and axle, and inclinde plane being the other five. It is primarily used as a force multiplier {i.e., you apply a small force called EFFORT (E) on one side to lift a larger force called LOAD (L) at some other side}. Machines make your tasks easier.

a. You would use a single fixed pulley to make your E easier. It is easier to pull something with gravity rather than against gravity. What you have done is instead of struggling to lift something UP, you pull DOWN on the rope of a single fixed pulley. All that you have achieved is a change in the DIRECTION of the force.

Ideally E = L, and therefore the mechanical advantage (M.A.) = 1. In reality you would always lose some energy in overcoming friction in the bearings of the pulley.

b. You would use a single movable when you want to apply less effort E. To lift a load L you only have to apply an E = L/2. This is fine as long as you are willing to pull the object UP.

Assuming like above no energy losses, the M.A. of a single movable pulley would be b.

Obviously you're likely to couple this with a single fixed pulley to make it earier to change the direction of the force.

Our ancients were smart too and they were quick to realize that although you gain in force, you lose in distance. That is, when you apply less effort E, you are invariable applying the force over a larger distance. So you gain force but lose distance! You can't have it all, can you?

5.    This is one of the quirks of physics. The answer is yes in the physicists lingo. In physics we say work is said to be done when a force moves an object IN the direction of the force. It is calculated as you have correctly done by multiplying force and distance and measured in Joules.

Experience tells you and me that we do feel tired when you push a heavier load than a light one as you've correctly observed. This is somewhat similar to what you do in the Gym with different loads. Lesser the load easier it is to move.

The catch here is there are two DIFFERENT processes taking place. What the physicist is describing is the external work done by a person on an object. This could very well have been done by a machine as well. This problem is similar to running up a flight of stairs, going on a ramp and trying to climb VERTICALLY UP. What you and I indeed gain here is TIME.

Coming back to our problem in order for us to accomplish this change our bodies have to provide for this energy by chemical reactions taking place inside the body. They've got to be continually supplying sufficient energy to move these loads. There is only so much our cells can do, although one can train them to supply more (gym agian!) through training.

Although we do get tired moving the heavier load, the external work done according to the physicist is still the same.

6.   (i) Perpetual motion machines have captured the imagination of scientists and artists for a very long time. It is basically any device which can generate more energy than it consumes or a device which can keep working forever. As a physics teacher it is always a pleasure to interact with several school students on this topic. The consequences of this if we consider the human machine is significant as well!

Whenever I speak of the 'Energy Crisis' year after year there is at least one student who wants to obtain 'free energy' with his/her ingenious design. You can see this evidenced on some of the links I have suggested below, showing the interest of scientists as well. I was always fascinated with Escher's paintings and one of my favorite sketches can be found on the URL below with Escher's impossible buildings.

    http://www.worldofescher.com/gallery/Waterfall.html

To see numerous designs of such machines and useful links, you can visit Eric's History of Perpetual Motion and Free Energy Machines

    http://www.phact.org/e/dennis4.html

The Perpetual Motion Page has a few designs for perpetual motion machines as well including one of my favorite Nobel Prize winning physicists' R. P. Feynman.

    http://manor.york.ac.uk/htdocs/perpetual/perpetual.html

(ii)    I think you'll find the answer to the question "Is perpetual motion possible? If not, why?" at the URL below interesing.

    http://www.madsci.org/posts/archives/sep99/936709021.Ph.r.html

I'm also suggest you read my answer to the question on superconducting magnet, under electric circuits, interesting reading.

(iii)    May be there can be no correct answer as yet? There are various "theories" to explain why the core is so hot still. Yes nuclear reactions or more precisely radioactive decay is certainly believed to be a major cause. The other theory believes that this is the heat left over from the time when the earth was formed billions of years ago.

All of us know that any object hot must start losing energy. Therefore, the earth's core must certainly be cooling off too. If you're wondering why it has not cooled off completely as yet, well there can be several factors. One of the main factors would be the miles of rocks in the mantle and crust acting as good insulators of heat. The other could be the radioactive decay constantly generating more heat.

If you want to tap that energy, you have to contend with both the rocks and radioactivity. Are we ready yet?