## Comparison of Relative Energies and Powers

A lot of times, you can evaluate a statement or theory by just looking to see if the energy and power involved are reasonable. To help you out, here is a list of the energy and power (rate of energy) for some common and not-so-common things. There has been a lot of judicious rounding and estimation in this table, so don't use it for precise quantitative work. Use it to verify stories and claims to an order of magnitude.

The energy is given in the standard SI unit of Joules. A Joule is the same as a newton meter. Mechanical energy is usually given in Nm, and electrical and heat is usually given in Joules, but they are really the same unit. A watt is one Joule in one second, or to put it another way, a Joule is a watt-second. For comparison, a Joule is about .74 Foot Pound. A horsepower is 550 ftlb/second or 745 watts.

1 Newton Meter = 1 Joule =  .74 FtLb

I have included some discussions of notorious, and inaccurate, comparisons. These are designed to show that mere comparisons of energy aren't necessarily significant, and that a little qualitative analysis helps to separate the marketing hype from reasonable facts.

### Comparative Energies Table

 Energy (Joules - NewtonMeters) Example Power (Watts) 1 Nm An apple falling from 1 meter 40W (hitting table and stopping in .025 seconds) 230 Nm Fastball @ 100 mi/hr 2300 W (caught in mitt in 0.1 seconds) 450 Nm .357 Mag Handgun (150 gr @ 1000 fps) 12.5 MW (impact into steel plate) 3900 Nm 7.62x39 Rifle Bullet (120 gr @ 3000 fps) 325 MW (impact into steel plate) 5000 Nm 220 lb tackle running 40 yd in 4 sec (100 kg at 10 m/sec) 20 kW (assuming .25 second hit) 490 KJ Kinetic energy of Automobile (3000 lb @ 60 mi/hr) 179KW (panic stop at 1 G) or 2.1 MW (crashing into concrete wall) 2 MJ 1 Lb High Explosive 240 GW (detonation at 6000 m/sec) 4.32 MJ Energy in fully charged car battery 60 Watts (20 hr rate) or 4800 Watts (when starting car) 2.15 MJ Energy to run a car for 1 Mile (assuming 20 mpg & 33% efficiency) 1.2 kW (assuming car goes 40 mi/hr) 8 MJ Cup of Gasoline 3.2 GW (explosive mixture in air) 3.6 GJ Electricity I use in a month (1000 kWh) 1.4KW (avg) 80 TJ 20 kT nuclear explosion 8E19 W

#### The nonsense about gasoline and dynamite

A lot of times, it is the rate at which the energy is transferred or released that is more important than the amount of energy. There is an often repeated statement that a cup of gasoline is the same as 16 sticks of dynamite. Are they talking about the energy or the power (rate of release)? First off, energy probably isn't an appropriate comparison. After all, the energy in that cup of gas (around 8MJ) is the same as the energy in about a pound and a quarter of wood, which clearly isn't really comparable. Anyway, on an energy basis, the cup is more like 4 pounds worth of high explosive(8 MJ for the cup of gas, 2 MJ for the pound of HE).

More significant though, is that the explosive releases all of the energy in around 4 microseconds: a rate of 2000 GigaWatts. The gas-air mixture takes 2 or 3 milliseconds to explode (the combustion wave is limited to the speed of sound), for a power of around 3 GW. And of course, the wood takes several minutes to burn, for a power of around 30 kW.

For an interesting exercise compare this to a small nuclear explosion of 20 kilotons (= 80 Tera Joules = 8E13 Joules), where virtually all the energy release takes place in about a microsecond for a power of almost 10 to the 19 watts. 80 TJ is a bunch of energy (about what a 1000 MW power plant puts out in a day), but it is the phenomenally high rate that makes nuclear explosives really different.

##### Calculation background:

Gasoline has an combustion energy of about 47.4 MJ/kg, so a simple calculation will show that the cup (170g) will release about 8MJ when burned. When vaporized and mixed with air at the optimum ratio (stoichiometric), the mixture will occupy about 2100 liters, or a sphere some 0.80 m in radius. Assuming the mixture is ignited at the center of the sphere, it will take about 2.5 milliseconds for the flame front to propagate to the outer edge. This makes the assumptions that the flame front moves at the speed of sound (probably reasonable), and that the sphere of mixture doesn't expand appreciably due to the heating in the middle (probably very optimistic). So, 8 MJ in .0025 seconds is 3200 MW, or 3.2 GW

The detonation wave in most high explosives moves at around 4500 m/sec, substantially above the speed of sound (which is what makes it a detonation, and not just burning fast). That 4 pounds of explosive would be a cylinder about 5 cm in diameter and 20 cm long. At 4500 m/sec, it takes about 4 microseconds for the detonation wave to move through the entire amount. 8 MJ in 4 microseconds is 2000 GW.

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#### Regenerative braking is a marketing ploy

Another interesting observation has to do with the energy required to accelerate and run a car down the road. Much has been made of the ability to do regenerative braking with electric cars; that is, as you slow down the kinetic energy of the car is used to charge the batteries. For cars that run on the freeway, though, this is almost insignificant. The kinetic energy stored in the car moving at 60 mi/hr is about 500 kJ. That is the same energy it takes to go about a quarter of a mile (mostly overcoming air drag). Given the difficulty of doing regenerative braking for negligible increased range, you can see that regenerative braking is mostly a marketing ploy.

The problem with regenerative braking is that you have to charge the batteries at a very high rate (much higher than the normal discharge rate) which makes them hot and have shorter lives. Also, the motor in the car, acting as a generator during regenerative braking, has to be big enough to handle the power. Our hypothetical car may only require a 10 kW motor to accelerate and drive around, but would require a 180 kW generator to decelerate at 1 G. Since the size, weight, and price of electric motors/generators runs as a linear function of capacity, you can see that regenerative braking costs a lot, but doesn't buy you much in range.

These same little facts apply to any scheme where you are planning on saving the energy in the vehicle, whether you are using batteries or flywheels or fuel cells, or whatever. You still need a big motor/generator, etc. There are applications which are limited to very low speeds where aerodynamic drag is insignificant, and the maximum deceleration is limited, like urban transit buses. In these applications, regenerative braking does work, just not for cars on the freeway.

Another interesting automotive calculation is the idea of covering your car with solar cells. As a rough approximation, the average power of the sun falling on a flat surface is about 1kw/square meter. If your car was 4 meters long and 2 meters wide, you'd be collecting 12 kW (best possible scenario with perfect solar cells) (about 16 HP). With real solar cells (10% efficiency), and taking into account the sun being at an angle, you're looking at around a kilowatt (just over a horsepower), which might just move your moped around town, but certainly not a car on the freeway. The power requirement goes up as the cube of the airspeed, so doubling your speed takes 8 times the kW (or HP).

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#### Knocking down the bad guy with a shot to the hand

This table also illustrates the fallacy of movie lore of a man being shot (particularly in an extremity) and falling down from the impact. The energy in a .357 Magnum handgun round ( a fairly high power example) is only about twice that in a fastball at 100 mi/hr. You can get hit by a baseball (particularly in the hand) and not be knocked down (although it will hurt a bunch), and likewise, you can get hit by a bullet and not get knocked down (although it will hurt a whole lot more). The FBI did a study on this a few years back and concluded that in most cases, the victim falls down because they are surprised or because they think, subconsciously, that they should fall down when shot. When the victim doesn't feel the pain (i.e. they're drunk, drugged, or very motivated), they don't fall down! I leave aside here the question of injuring someone in a vital area and killing them, which will make them fall down, whether by baseball, bullet, or fast moving hand.

Also notice that the rifle bullet or football tackle, with energies of several kJ, will knock you down.

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