Re: Advice about illegal fireworks
This will explain it all about a projectile fired upwards.
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Vertically Shot Bullet Landing Speed
name Davidstatus educatorgrade 9-12location PAQuestion - In the real world, (not the physics world of no air resistance), how do I calculate the speed of a bullet that was shot straight into the air when it returns to Earth. For example, say a rifle had a muzzle velocity of an M-16, what would the bullet's speed when it comes straight back down. On the news one sees celebrations in the Middle East of shooting into the air. Just how dangerous is that?--------------------------------------------------------------
The upper limit is of course as you point out (no air resistance). Including air resistance is a much more complex calculation because it depends upon many other factors -- air density as a function of height, the shape of the bullet, the rotational speed of the bullet, whether the bullet is wobbling or tumbling in the way down Really a complicated mess to calculate, but for the moment DO ignore air resistance. A rule of thumb in ALL physics/chemistry/engineering is do an order of magnitude calculation to see what, if any, more complicated calculation may be necessary -- but always carry out "reality checks" to make sure you're on track. Let's consider a rifle bullet vs. a hailstone (spherical). RIFLE BULLET: A typical muzzle velocity of a rifle (google search) is 3000 ft/sec = 1000 m/sec (notice I'm rounding here because we are just looking for where the decimal falls.A typical bullet mass is 120 grains [wierd units, but 1 grain = 0.065 gm] = 7.8 gm = 10 gm (close enough). Now a "reality check". The density of lead is 11.4 gm/cm^3 = 10 gm/cm^3 (close enough). So the volume of the bullet is: volume = mass / density = 10 gm / 10 (gm/cm^3) = 1 cm^3. That's probably pretty conservative, but OK for an order of magnitude. Remember the shell casing doesn't count -- only the projectile. In the absence of air and a perfect world, kinetic energy is conserved, so the bullet weighing 10 gm will hit the ground after a vertical trajectory at a speed of 1000 m/sec. OUCH!!! Let's calculate the energy. From the muzzle velocity (1000 m/sec) and the bullet mass (10 gm = 10^-2 kg) and K.E. = 1/2 m(v)^2 we get 1/2 (10^-2)*(1000)^2 = 0.5*10^-2+6 = 5x10^3 = 5000 Joules. HAILSTONE: A spherical hailstone weighing 10 gm has a volume of 10 cm^3 since the density is1 gm/cm^3. REALITY CHECK: The volume = 10 cm^3 = 4/3 *pi* r^3. So r^3 = 2.4 cm^3 or a radius of r = 1.3 cm or a diameter of 2.6 cm (a fairly nominal hailstone -- about an inch in diameter).The potential energy of a hailstone weighing 10 gm = 10^-2 kg falling from 10 km is: P.E. = m*g*h =10^-2 * 9.8 * 10 (about 10^-2+1+1 = 10^0 = 1 Joule). Its velocity assuming complete conversion of the P.E. to K.E. = 1/2*m*(v)^2 gives: 1 = 1/2x10^-2*(v)^2 or v^2 = 200 m^2 or about 15 m / secCompared to 1000 m/sec for the bullet. Now if you want to refine the estimate further a fair assumption would be to assume that air resistance would be proportional to the cross sectional area of the object and the time of flight. That is, the longer the object is in the air the greater will be the drag from the atmosphere until the projectile reaches its maximum terminal velocity. Even without doing the calculation, assuming the bullet is a cylinder (you can vary the length / diameter ratio) and the hailstone is spherical, the air resistance will be much less for the bullet than for the hailstone for two reasons -- cross sectional area, and time of flight. This also gets a bit messy because the bullet experiences drag both going up and down, but the hailstone only experiences drag on the way down. Of course, there are other complicating factors that have been ignored. A big one is that the speed of the hailstone will depend upon whether it is falling in an up-draft, or is being accelerated by being in a down-draft.
As to danger there is no question that getting hit by such a spent projectile could be lethal. Actually getting hit by a 1 inch hailstone would be very unpleasant and possibly also fatal.
Vince Calder
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Talking about making my head hurt.
