Handout #3

ASTR220 – Spring 03 – Dr. Reynolds

(originally produced by  Dr. Harris)

 

 

Impacts and energy

What makes an impact with a mountain-size asteroid so much more destructive than the same size mountain sitting on the Earth’s surface?  The difference is the relative speed of the asteroid and the Earth’s surface; while one of the Earth’s mountains sits placidly on the surface, an asteroid may well be moving toward the Earth’s surface at a speed of 50,000 to 100,000 km per hour (5´104 to 105 kph).  Compare this to a brisk highway speed of 110 kph (70 mph), and you get a sense of how fast a colliding object can move.

 

You know from experience that there is energy in motion – the faster you go, the harder it is to slow down and the more painful it is if you hit something before you can stop.  This energy of motion is called kinetic energy.  Mathematically, the amount of energy is related to the mass and velocity of an object by:

                                                                                                                        

where m is the mass of an object (we’ll use kilograms, or kg) and v is the velocity (we’ll use meters per second, or m/s, or m s-1).  With these units, the energy E has dimensions kg m s-1, otherwise known as Joules, with symbol J.  What does this equation mean?  It tells us that the amount of energy is directly proportional to mass (twice the mass, twice the energy) and is proportional to the square of the object’s velocity (twice the velocity, four times the energy). 

Understanding velocity is straightforward: it is simply the rate at which something covers distance.  A very familiar example is speed in miles per hour.   (Velocity and speed are closely related: technically, velocity includes both speed and direction, for instance 10 mph toward the north, while speed is only the 10 mph part without any directional information.)  In more usual units for scientific use, we’d think of meters per second, but you can see that the idea remains the same: it’s just distance divided by time.

Mass is also something we’re familiar with but usually don’t think about very carefully.  We generally use mass and weight rather interchangeably; for instance, it’s common to ask to buy, say, 2 kg of apples.  If you took your 2 kg of apples into orbit around the Earth, you know that the apples would be weightless, but that you’d still have 2 kg of apples; and on the Moon, where gravity is weaker, you’d have lighter apples but still 2 kg.  Weight, when you think of it, is really a force, but mass is a property of matter that is unaffected by the gravity it finds itself in.   The reason for our confusion between mass and weight is that scales on the Earth measure the force between the mass of the Earth and the mass of the apples, with the numbers on the scale set up for exactly this case.

Density is another intrinsic property of a material: it is the mass per volume, with units of kg per cubic meter, or kg/m3, or kg m-3.   Water, for instance, has a density of 1000 kg per cubic meter, and that’s true whether you have a cupful or an oceanful.  A cup has much less volume (cubic meters) than an ocean, of course, so an ocean has much more mass.  Two of the impact experiments involve balls of different sizes and different materials.  The mass of the steel balls depends on their size alone – the density of the ball material is always the same – but the mass of the rubber and foil balls can be different even for the same size balls since the densities are different.  The mass m (units kg), density r (units kg/m3), and volume V (units m3) are linked by the expression

                                                                                                                            

You see that the units work out fine when you multiply them in the same way: kg = kg/m3 m3, with the m3 term from volume canceling the per m3 term in density.  You’ll remember from high-school geometry that the volume of a sphere is                                                              

                                                               ,                                                         

where r is the radius of the sphere (units m), and the 4/3 and p (3.1415926…) are constant numbers.  What this expression says is that the volume of a sphere is proportional to the cube of the radius (twice the radius, eight times the volume). 

When we combine equations and , we get the expression

                                                              ;                                                        

that is, the mass of a sphere is proportional to its density and to the cube of its radius.  Making one more step by combining equations and , we get a compact expression for the energy of an spherical impactor:

                                                 ,                                           

where I’ve just grouped all the constant terms together in the last part of the equation.  If the object isn’t a sphere the numbers up front (4p/6) will change, but otherwise expression remains unchanged (density is density, volume is volume, and velocity is velocity).

 

Energy and impacts

Expression tells us some interesting things in a compact way.  The energy of an impact will be proportional to:

·       The density (twice the density, twice the energy)

·       The cube of the radius (twice the radius, eight times the energy)

·       The square of the velocity (twice the velocity, four times the energy)

So, in terms of something hitting us, what should we worry about?  The most dangerous objects will be dense, large (lots of volume), and moving fast.  Given a choice between being hit by an iron asteroid or a comet of the same size, moving at the same velocity, we’d be happier with the comet: the density of ice is considerably lower than the density of iron.   (An iron cube would be a lot heavier than an ice cube of the same size!). 

The fact that, all else equal, the energy increases as the cube of the radius means that we should be very alarmed by large objects.  An object that is twice the diameter of another will have 23 = 2´2´2 = 8 times more energy, all else being equal.  For three times the diameter, the factor is 27, four times 64, five times 125, and so on.

The situation is only slightly better for velocity: we’d much prefer to be hit by something moving slowly than something moving quickly.  (You know this from experience: you wouldn’t want to catch a softball that’s been fired from a cannon.)

 

 

Conservation of Energy

A fundamental rule in physics is conservation of energy: energy can change from one form to another but can’t be destroyed.   Energy can swap from kinetic energy to heat to light to potential energy in a gravitational field, but at any instant a detailed accounting will show that the total amount of energy (measured in Joules) is the same as always.  It’s kind of like sharing water between many cups and barrels: however you divide the water between all the containers, the total amount is the same as long as you don’t spill any.

 

A nice example of swapping one kind of energy to another is the transformation of gravitational potential energy into kinetic energy when you drop a ball as part of the cratering experiment.  The expression for gravitational potential energy is

                                                                                                                            

where E is energy, still with units of Joules; m is still mass, units of kg; g is the gravitational acceleration, which is 9.8 m/s2 near the Earth’s surface; and h is the height in meters. 

We can balance the energy in expressions and ; E = E, or

                                                             .                                                       

We could use this to solve for the velocity (the m on each side cancels out, then move the 2 and take the square root of each side),

                                                                                                                          

or we could directly see that the ball will gain energy mgh when it falls through a distance h.  This means, if we just drop the ball and don’t throw it, the energy of the impact should increase directly with mass m and fall height h.  In terms of density and volume, the total energy in our experiment impact will be

                                                 .                                           

We’ll expect to see the energy change

·       Directly with density (twice the density, twice the energy)

·       As the cube of radius (twice the radius, eight times the energy)

·       Directly with height (twice the height, twice the energy)

If we could do the experiment on the Moon, where g is smaller, we could see that too, but as long as we stay on the Earth’s surface we can’t explore changes in g.

 

Energy in a real collision

So what happens to energy in a collision?  In most collisions, kinetic energy changes to heat.  What’s heat?  Mostly kinetic energy of atoms and molecules.   In our cratering experiment, the ball falls into the flour and throws the flour around.  In a cosmic impact, where the velocities are much, much higher (and remember that the energy depends on the square of the velocity: a velocity a thousand times higher than our experiment means a million times more energy for the same impactor), the impactor will trade its kinetic energy for so much heat that it and much of what it hits will vaporize.  (Vaporize: the atoms of the solid start moving quickly enough that they first melt and then become a gas.)  The material in the impact will get so hot that it glows like a very hot version of the burner on an electric stove.  An enormous amount of energy is converted from the energy of motion in one big thing to energy of motion (heat) and light in many small things.  We sometimes think of this transformation inaccurately as a “release” of energy.  The effect of dumping this huge amount of energy into formerly calm, quiescent material is exactly the same as a bomb (which releases a huge amount of energy into formerly calm, quiescent material).