NEWTON’S LAWS AT WORK: Working Out The Energy Of An Asteroid.

over 4 years ago

It is very rare for an asteroid or a comet to hit the Earth. But smaller pieces of matter arrive from space all the time. Each day, 400 tonnes is added to the earth’s mass from all the pieces of metal, rock and ice travelling through ‘empty’ space which are trapped by the Earth’s gravity and fall into its atmosphere.

^{A meteorite: Pixabay.}

Meteorites chunks of matter in space – fall very fast, between 10 and 70 kilometres per second. Friction with the atmosphere heats them to extremely high temperatures, and that is why we can see them at night as ‘shooting stars’. Even objects of 50 metres in diameter generate enough heat to melt and vaporize completely in the upper atmosphere. This is just as well, as an object of that size has the kinetic energy of a 10-megaton nuclear bomb (4.2 × 10¹⁶ J).

Massive objects have collided with the Earth about once in every 100 million years. The Earth and the other inner planets of the Solar System would have experienced many more collisions but for the giant planets Jupiter, Saturn, Uranus and Neptune, whose gravitational pull tends to divert the path of large objects. In 1994, for example, fragments of Comet Shoemaker-Levy crashed into Jupiter. It is estimated that, without the giant planets, similar collisions with the Earth could rise to once every 100,000 years. Currently, groups of scientists have been doing great work on ways of detecting any large objects likely to hit the Earth – and what might be done about it if any are detected.

Energy on a massive scale

The Solar System is an arrangement of planets in orderly orbits round the Sun. But in addition, countless objects smaller than the planets take other paths, and sometimes collide with the planets. They include, for example, asteroids, and meteorites, which are smaller bodies. Near-Earth asteroids are straying members of the asteroid belt, a mass of material that lies in a wide band between the orbits of Mars and Jupiter. There are at least 1800 large asteroids with a diameter greater than a kilometre that cross the Earth’s orbit – and so are possible candidates for a future catastrophic collision. The figure below shows the asteroid Eros, on which the space probe NEAR (Near Earth Asteroid Rendezvous) actually landed in February 2000.

^{asteroid Eros. NASA/JPL/JHUAPL, public domain}

Meteorites are smaller, mostly fragments of comets, consisting of metal and rock dust embedded in frozen water and gases, which come from the Oort cloud, on the outermost fringes of the Solar System. An asteroid hitting the ground makes a crater far larger than itself, because of the colossal energy of its impact. An example is the 180-kilometre-wide crater at Chicxulub, Yucatan, in Mexico. The asteroid is thought to have been just over 10¹² tonnes in mass and 10 kilometres in diameter. It collided with the Earth 65 million years ago, at the end of the geological period known as the Cretaceous, and with an estimated kinetic energy of about 4 × 10²³ J. This is about a thousand times greater than the total annual global energy used by human beings today (4 x 10²⁰ J).

The asteroid impact at Chicxulub caused a major upheaval to life on Earth. Such a massive impact sends volumes of dust into the atmosphere, which circulates round the globe for years and cuts out sunlight. Plants fail to grow and animals that depend on them die. The decline of the dinosaurs is the best known effect thought to have been caused by the Chicxulub impact, but it is thought that about half of all marine species died out at the same time.

Similarly, 160 million years earlier, 95 per cent of sea life disappeared, probably as a result of a larger impact by an asteroid whose remains have been found in the South Atlantic Ocean.

The ideas in this post…………

The physics: To work out the energy of an asteroid, you need to understand the ideas of kinetic energy, gravitational potential energy and also the principle of the conservation of energy. You will also be reminded about power as rate of working or transferring energy. You will develop the notion of a field of gravitational force obeying the inverse square law. You will learn how these forces and fields determine the orbital motion of asteroids, satellites and planets. You will also meet the physical principles of how rockets can put satellites into orbit, and so gain a good understanding of momentum and its conservation.

The mathematics: On the whole, simple algebra is all you need. You will be handling large numbers, interpreting graphs and learning of the importance of the area under a graph. Though not essential, integral calculus and the ability to use a spreadsheet would be helpful as well.

ENERGY AND WORK

I have discussed the consequences of an asteroid, an object of massive energy, hitting the Earth. Now, we turn to the concept of energy on a very much smaller, everyday level. We base our idea of energy on the concept of work, since the energy of a system is both defined and measured in terms of the work a system can do. The word ‘work’ is defined by physicists to mean something more precise than its common use, because we say that:

Work is done only when a force moves something

For example, work is done:

when you use muscular force to lift an object upwards against the gravitational force
when you saw through a piece of wood, again using muscular force to tear the fibres of the wood apart
when an electrical force moves electrons through a resistor
when water under pressure turns a turbine in a hydroelectric power station
when the petrol-air mixture in the cylinder of a car engine explodes and pushes the piston outwards.

^{Loading a truck on a ship using a ramp. U.S. Navy photo by Paul Farley, Public Domain}

Forces doing no work

You are not doing any work when you stand still with a heavy rucksack on your back. You get tired because your muscles are stressed. But if the rucksack stays still, no work will be done on it. In the same way, a table wouldn’t be doing any work if you took a rest and put your rucksack on it.

THE WORK FORMULA

The quantity of work done by a force is measured in joules:

Work done (in joules) = Force (in newtons) × distance moved in the direction of action of the force (in metres)

W = Fd

The direction of the force F, and the movement of the point where the force is applied, have to be taken into account.

EXAMPLE

A man pushes a shopping trolley up a ramp 5 m long. He applies a force of 12 N along the ramp. How much work has been done?

Solution: Work done (J) = force (N) × distance moved in the direction of motion (m)

= 12 N × 5 m
= 60 Nm (60 J)

POWER

It has been calculated that the average citizen of Britain has at his or her disposal the power that only the richest Ancient Romans might have had – equivalent to several hundred slaves. Power is a measure of how quickly work is done, and so of how quickly energy can be transferred:

Power = rate of working = energy transferred / time taken

P = E / t

Power is measured in joules per second and has its own special unit: the watt (W).

In the example above if the man pushing the trolley up the slope did the work in 8 seconds he would be working at the rate of 60 J / 8 s or 7.5 W

ENERGY IN A GRAVITATIONAL FIELD

Because work is usually easy to measure, we often describe the energy of a system in terms of how much work it does – or is capable of doing. Then we can define energy in this way:

A system has energy if it is capable of doing work.

GRAVITATIONAL POTENTIAL ENERGY

When you lift something, you are doing work. Think of taking a bag of sugar from the floor and putting it on a shelf. To do this, you have to exert a force that is just greater than the force of gravity on the sugar bag (its weight, mg), and you have to raise the bag through a height, h.

The work done on the bag = force × distance

W= mgh (in joules) or W = mgΔh

For a change in height Δh.

In a system of two objects, gravity provides a pair of forces of attraction that are equal in size but opposite in direction – the Earth pulls on the bag and the bag pulls on the Earth. If the sugar bag fell off the shelf, both bag and Earth would move towards each other (though the Earth would move only a tiny distance).

The sugar bag could do a little useful work while it is falling – for example, we could make it briefly operate a very small generator and produce an electric current to light a small bulb! On a much larger scale, the waterfalls are used to produce electricity in hydroelectric power stations.

So we can think of the Earth-sugar bag system as having ‘stored’ energy when the bag is resting on the shelf. We call this energy gravitational potential energy. The word potential reflects the fact that the sugar bag doesn’t look very energetic when it is just sitting on the shelf. But we know that it is capable of doing work when it falls. We have noted that work is done when a force moves something. If we don’t use the gravitational force on the bag to do useful work (such as turning a tiny generator), then the work done simply increases the kinetic energy – the ‘energy of movement’ – of the bag: as it falls, it moves faster. The energies in their different forms are equal:

Potential energy → kinetic energy

The mass m picks up speed v so that ½ mv² = mgh. This is a very useful relationship.

THE PRINCIPLE OF THE CONSERVATION OF ENERGY

It is not obvious that the work done in lifting the sugar bag to the shelf is equal to the potential energy stored when it gets there, and that this potential energy is equal to the kinetic energy it would gain as it fell back to the floor.

In any case, you can’t actually measure the potential energy directly. If you carried out a careful experiment to measure the work done and the final kinetic energy acquired, your results would probably show that the work done and the final energy were only approximately equal. It is, for example, hard to allow for work done against friction, or for experimental errors.

Nevertheless, one of the most fundamental principles of science states that: energy cannot be created or destroyed.

Belief in this principle grew stronger during the nineteenth century as a result of many increasingly accurate experiments, and scientists gradually came to accept it. It became known as the principle of the conservation of energy.

EXAMPLE

A rock of mass 25 kg falls from a clilf to a beach 30 metres below. What is

a) its kinetic energy

b) its speed just before it hits the beach?

Solution: Relative to the beach, the gravitational potential energy of the rock before falling can be expressed as:

mgΔh = 25 × 9.8 x 30 (in joules)

= 7350 J

a. Energy is conserved so, ignoring friction with the air, the kinetic energy of the falling rock is also 7350 J

b. Therefore:

½ mv²= ½ × 25 x v² = 7350 J

v²= 7350 × 2/25

v²= 2 × 7350 / 25

v = 24 ms^-1

Speed of rock just before it hits the beach = 24 ms^-1.

The kinetic energy formula

I have already used the formula for kinetic energy above and it is:

Kinetic energy = ½ mv²

which describes the kinetic energy of an object in terms of its mass and speed. We can derive this formula by looking at the kinetic energy of an object on which work is done.

^{The cars of a roller coaster reach their maximum kinetic energy when at the bottom of the path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy. The sum of kinetic and potential energy in the system remains constant, ignoring losses to friction. Wikipedia, CC BY-SA 3.}0

Let’s assume a car having work done on it, and gaining speed and so kinetic energy. A net accelerating force F acts on the car, mass m, and the car moves a distance x from rest, gaining speed, v.

According to the principle of conservation of energy, the work done, Fx, is transferred to the kinetic energy of the car (ignoring friction and other energy losses):

Work done = force × distance moved = Fx = max

(since force F = mass × acceleration). It is this work that has given the car its kinetic energy. We now relate ax to speed v using the kinematic equation of motion, v²= 2ax, for an object starting at zero speed

Rearranging this equation,

ax = ½ v²

So we can write:

Kinetic energy gained = m(ax) = m × ½ v²

Which we normally write as:

E_k = ½ mv²

I’ll be pausing here for now, so that the post won’t be too lengthy. In my next post, I’ll discuss further on Joule’s work on heat and energy, energy conservation in a uniform gravitational field, gravitational potential energy in the earth’s field and the negative energy. But, till then I still remain my humble self, @emperorhassy.