Simple yet compelling examples are rare. Here we consider the momentums of two entities.

P-51 MUSTANG On straffing runs WWII P-51 Mustangs often fired bursts of 1000 rounds from its six - 50 calibre machine guns. Those streams of fired lead changed the air speed (momentum) of the Mustang. Calculate the P-51's air speed after it fires a 1000-round burst.

We take the system to be the mass of the P-51 and the 1000 rounds of 50 cal bullets.The mass of such fighters was 5400 pounds and its speed 300 miles per hour (440 ft/s) when firing commenced. Each bullet (mass = 0.1 lbm) left its machine gun muzzle with a speed of about 3000 feet per second. A sketch of before and after is helpful.

We take the airplane and the bullets it carries (then fires in the event) as our system. Since there is "plane" momentum and "bullets" momentum, we place a summation, left of equality, on momentum.

An event is a change from some initial condition to a final condition. Initally the plane and bullets move horizontally at constant speed. The state is called uniform motion. Propeller thrust causes the velocity, the drag (the opposing force of air as the plane moves through it) equals the thrust force. So velocity and momentum are constant for all times prior to firing (t < 0+). Stated by equation, this is:

The above equation tells that momentum is zero (in two ways) prior to firing. The six guns are triggered in a burst at t = 0+. To see what the equations become, we write the momentum equation as:

The above equation is complete. You might wonder about forces as the guns recoil. But the recoils are not forces because the bullets and aircraft are both part of the system. Forces of the momentum equation are effects of the surroundings. Headed toward a solution, we separate variables then apply the integral:

We like to do easy parts first. The integrand, right of equality (F_thrust - F_propellor) was zero before change started, when time was t < 0+. We suppose the firing happens very fast so that either "dt" is zero or (F_thrust - F_propellor) remains what it was to start, zero. Use either approximation to set "right of equality" equal to zero for the event. Left of equality integrates immediately.

The summation was required - our system has two components. Now we expand the sum.

The direction of all vectors is to the right, so we scalar multiply or "dot" multiply ("• I") the equation then put numberes into it.

This amounts to a worrisome, 14% decrease in speed. We suppose 14% is the least reduction. In events of "squandered energy," approximations tend to be "rosey."

So we solved the "P-51 and Bullets" first order differential equation, in the perfunctory, classical way. Separate variables, set limits, approximate and integrate. Hopefully you have solved a first order differential equation before. If not, no matter. This very same thing is done, the very same way, more than a dozen more times in the next pages. First order differential equations explain physical reality with mathematically generally. To be specific, set the initial conditions and integrate (or approximate). We must understand this powerful connection of calculus with physics and event.