Collisions in Space Handout #3
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 speed of an object by:
E
1 2
mv
2
(1)
where m is the mass of an object (we’ll use kilograms, or kg) and v is the speed (we’ll use
meters per second, or m/s, or m s-1). With these units, the energy E has dimensions
kg m2 s-2, 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 means
twice the energy) and is proportional to the square of the object’s speed (twice the speed,
four times the energy).
Understanding speed is straightforward: it is simply the rate at which something covers distance. A very familiar example is speed in miles per hour. 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. The mass m (units kg), density  (units kg/m3), and volume V (units m3) are
linked by the expression
m  V
(2)
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
4
V   r3 ,
3
(3)
where r is the radius of the sphere (units m), and the 4/3 and  (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 means eight times the volume).
When we combine equations (2) and (3), we get the expression
4
m    r3 ;
3
(4)
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 (4) and (1), we get a compact expression
for the energy of an spherical impactor:
E
1 4 3 2 4
 r v 
 r 3v 2 ,
2 3
6
(5)
where we’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 (4/6) will change, but otherwise expression (5) remains unchanged (density is density, volume is volume, and speed is speed).
Energy and impacts
Expression (5) 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 speed, 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.
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The situation is only slightly better for speed: 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. The expression for
gravitational potential energy is
E  mgh
(6)
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 (1) and (6); E = E, or
mgh 
1 2
mv .
2
(7)
We can use this to solve for the speed (the m on each side cancels out, then move the 2
and take the square root of each side),
v  2 gh
(8)
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 this case will be
4
4
E    r 3 gh 
g  r 3h .
3
3
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).
3
(9)
If we could do this experiment on the Moon, where g is smaller, we could see that the energy will also be smaller (all else being equal), but as long as we stay on the Earth’s surface we can’t easily 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 a cratering experiment like dropping a ball into flour, the ball throws the flour around. In a cosmic impact,
where the speeds are much, much higher (and remember that the energy depends on the
square of the speed: a speed a thousand times higher than our flour 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” means
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).
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