The trajectories of projectiles have long been of practical interest, and the study of drag shows how to apply effective streamlining to cars, boats, trains and planes.

The first internal combustion engine was the gun, its piston the projectile that left its muzzle, propelled by the gases at high pressure produced by the explosion of its charge. The intention of the gunner was to send the projectile to a predetermined point, to execute its destructive or evil purpose. The importance of this process in the affairs of mankind has led to the study of the paths of projectiles, or *ballistics*. At school, ballistics is a problem in uniformly accelerated motion, and the trajectory is a parabola, determined by the initial velocity and the angle of projection. In practice, this description falls far short of reality because of the influence of the atmosphere on the motion of the projectile. In studying the consequences of air resistance, we will come across many peculiar facts, and expert guidance in the matter of streamlining.

Suppose that we measure the velocity of a projectile in level flight at two points separated by a distance x, finding values V and v. The average retarding force at velocity (V + v)/2 is then F = m(V^{2} - v^{2})/2x, where m is the mass of the projectile. This force is called *drag*. If the drag as a function of the velocity is known, then the trajectory of the projectile can be calculated, at least in principle. We require, therefore a means of measuring the velocity of a projectile.

A means was devised by Benjamin Robins (1707-1751), the *ballistic pendulum*, around 1842. The projectile is fired into a heavy pendulum, to which it transfers its momentum. From the swing and weight of the pendulum, the velocity of the projectile can be found. This is still an experiment performed in the Physics laboratory. Robins measured the velocity of the projectile at various distances from the gun, using the same charge each time. He showed that the drag was roughly proportional to the square of the velocity, the first time this was done, but accurate results were impossible due to the lack of reproducibility of the muzzle velocity even when the same charge was used.

Hutton, at Woolwich Arsenal around 1812, suspended the gun as a pendulum itself. From its recoil, the muzzle velocity could be determined, so that accurate results for the drag as a function of velocity could be obtained. Bashforth, also at Woolwich, devised an electrical timer that gave the time at which the projectile passed through each in a sequence of screens. From these times, the speed as a function of distance can be determined. Around the time of the First World War, wind tunnel measurements began to measure the drag directly. So important were these measurements regarded that they were initially military secrets.

The drag on a projectile can be expressed as F = ρd^{2}v^{2}f(v/c, vd/ν). Here, ρ is the density of the air, d is the diameter of the projectile, v its velocity, and the dimensionless quantity f(M,R) is called the *drag coefficient*, depending on the dimensionless parameters M = v/c, the Mach number, and vd/ν, the Reynolds number, where ν is the kinematic viscosity of the air, about 0.144 cm^{2}/s under normal conditions. When comparing drag coefficients, carefully note the defining equations, which may differ in the constants used.

For values of R below some critical value, on the order of 100 or 1000, the drag coefficient f = C/R = Cν/vd, so F = Cρνdv, where C is some constant. The drag is proportional to the velocity, and to the linear dimension of the projectile. This *viscous* regime is rather unimportant for projectiles, since even for a projectile as small as 5 mm diameter, the critical speed is below 2.8 m/s.

Above the critical value of R, f is roughly constant, say f = C', and F = Cρd^{2}v^{2}, proportional to the square of the velocity and the cross-sectional area of the projectile. The nature of the flow is shown in the Figure. It is the same whether the projectile is considered to move in still air, or whether the air is moving past the fixed projectile at the same speed. Streamline flow occurs where the streamlines are converging. There is a stagnation point exactly at the nose of the projectile, whatever its shape, and otherwise the air flows smoothly around the obstruction. If there were no viscosity, the streamlines would close in again behind, and the air would exert no net force on the projectile: there would be no drag! Instead, flow in the diverging region is unstable, and a separation surface between streamline flow and the turbulent wake appears. The air in the wake has been accelerated in the direction of the motion, representing a loss of kinetic energy of the projectile, and this movement is dissipated in turbulent eddies. The pressure is low at the tail of the projectile, and this is the major contribution to the drag. A body that is not spherical tends to rotate so that its wake is as large in cross-section as possible, and the drag a maximum. This can be seen by dropping light objects, such as a leaf.

At around the speed of sound, Mach 1 or about 350 m/s, the drag increases abruptly, doubling or more, as we enter the transonic region, and then usually decreases slightly as the velocity is increased further. The reason for this change is a radical change in the flow around the projectile as M = 1 is passed. The new feature is the appearance of shock-wave cones at the nose and tail, making an angle Ω with the trajectory. The Mach number M = 1/sin Ω. These are sudden jumps in pressure, carrying considerable energy away, which explains the increased drag. The tail drag from the turbulent wake still exists, but it is dominated by the shock-wave dissipation. The pressure signature of the shock waves is a kind of N, which is heard as a distinct crack that appears to originate some distance behind the projectile. A strong shock wave travels at faster than the speed of sound, because of nonlinear propagation. The curve of the shock wave around the nose of the projectile is accompanied by a high pressure, which appears as drag. The idea is to create as weak a shock wave as possible to reduce drag, and this is best done with a finely pointed nose.

Although aircraft must travel at faster than 350 m/s to enter the supersonic regime, wave speeds on the surface of water are small enough that ships are strongly affected by wave drag. The bow wave is analogous to the shock wave of a projectile, and carries away energy that must be replaced by the propulsion system. There is also a turbulent wake, that sucks at the stern of the ship. The bows of the ship are pointed to reduce wave resistance. A canoe, which moves at less than the wave velocity is slender at each end to encourage streamline flow. Although there are obvious analogies, there are essential differences, such as the two-dimensional nature of waves on water and their dispersive propagation. The kinematic viscosity of water is only 0.01 cm^{2}/s, a factor of 10 less than that of air.

The streamlining of subsonic vehicles is a different matter, which should concentrate on encouraging streamline flow as far as possible, and reducing the wake to as small a cross-section as possible. The shape of the front is of little consequence, so long as it does not generate turbulence. Almost any form will create an invisible region of dead air ahead of it that acts as smooth fairing. The rounded fairing seen on the fronts of road trailers is almost useless in reducing drag; it would have a greater effect on the rear, where it might discourage separation. Any projections into the air stream create turbulence and consequently the drag. At the speed of automobiles, streamlining is only an esthetic matter. Indeed many of the 'fast' cars actually have shapes that increase drag, and appurtenances that are applied in a largely useless fashion. Any smooth outline will do as well as any other. On railways, the resistance to motion from other causes is so small that air resistance makes a considerable, even a major contribution, above 50 mph, and streamlining can be economical, a fact often overlooked. As in many other cases, additional power is more cheaply applied than streamlining. If you want a fast car, a big engine will do the job better than excellent streamlining.

When M > 1, the undisturbed air is separated from air influenced by the projectile by a strong shock wave. There is one shock cone at the front, and another at the rear, both compressive. The turbulent wake exists as before. If the shock wave curves around the nose, high pressures result that are chiefly responsible for the increase in drag. By making the nose sharp and pointed, this force is reduced. Making the tail slender has no effect, the turbulent wake being much as before. In contrast with subsonic streamlining, the body should now be slender in front, and blunt at the tail. This shows up in the shapes of subsonic and supersonic aircraft, where it is a critical matter to reduce drag as far as possible in either case.

The drag coefficient depends strongly on the angle between the axis and the direction of motion, called the angle of *yaw*. The drag coefficient also depends on the size of the projectile. It is greater per unit area for small cross sections than for large.

A projectile has an axis of symmetry, and the gun generally projects it so that this axis is parallel to the velocity as the projectile leaves the muzzle. This situation does not long remain, however, as a nonrotating projectile tumbles head over heels, greatly increasing the drag, and randomly deflecting the projectile to one side or another. Therefore, all projectiles were made spherical, the one shape that cannot tumble. There is, therefore, a very good reason why smoothbore guns fired only spherical bullets. An alternative is to provide fins or feathers at the back of the projectile to force alignment by aerodynamic forces.

It was known as early as 1500 that imparting a spin along the axis stabilized a projectile, and prevented it from tumbling. This was done by cutting spiral grooves in the barrel, or *rifling*. These grooves became fouled so rapidly that the advantages of rifling were doubtful. In addition, the bullet had to fit the barrel tightly, and this was also a drawback, especially when the gun was loaded from the muzzle. Although rifled hand weapons were successfully used by 1800, their general adoption had to wait for the development of non-fouling smokeless powder.

The adoption of rifling meant that projectiles could become lengthened, achieving a larger projectile for the same diameter of barrel or *calibre*. The increase to supersonic muzzle velocities favoured the sharp-nosed shape with only a slight contraction at the tail, the *boat-shaped* projectile. The nose of a projectile is usually *ogival*, that is, formed by rotating a circular arc tangent to the side of the projectile about the axis. The radius is specified in calibres (multiples of d). The nose may be rounded off by a spherical surface, also specified in calibres. When attempts were made to streamline the tail, it was found that such bullets tumbled and were useless. The shape of the bullet reflects its aerodynamics, not artistic design.

When the gun is considerably elevated, to achieve great range, or for the projectile to drop from above, the curved trajectory means that the axis of a spinning projectile cannot remain tangent to the trajectory, and complicated gyroscopic and drag effects enter. One result is *drift*, the departure of the trajectory from a plane, so that the projectile is deviated to right or left. Golfers, and others, are well aware of this phenomenon that accompanies a spinning ball. Mortar rounds, which are always used in high trajectories, have fins and do not rotate.

For the accurate computation of a trajectory, the forces on the projectile as a function of yaw angle must be known, and the projectile treated as a rotating rigid body. Then, with the drag known, the trajectory can be found by numerical integration, beginning with the initial velocity and spin. Not only is the problem a difficult and involved one, but the calculations are exceedingly tedious. In fact, they were so tedious that they stimulated the creation of digital computers in the 1940's, leading to the general-purpose computer, which has now revolutionized technology and life.

As an aside, the Reference mentions a "service time-fuze," which consists of a detonator pellet exploded by the shock of firing, and a train of gunpowder leading to the magazine of the fuze, of a length determined by the setting of the fuze. The problem of reliable time-delay fuzes is a difficult one. Note the spelling fuze, which is peculiar to ballistics. Such fuzes were used by scientists to determine the pressures acting on the nose of the projectile by the pressure effect on the rate of burning of the gunpowder.

F. R. W. Hunt, *The Reaction of the Air to Artillery Projectiles*, in Drysdale, et. al., *The Mechanical Properties of Fluids* (London: Blackie and Son, 1925) gives a clear introductory account.

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Composed by J. B. Calvert

Created 29 July 2000

Last revised