In guided air$-$to$-$air and surface$-$to$-$air rocket$-$propelled missiles the time of flight

to a given target, usually called the time to target $t_t,$ is an important flight performance

parameter. With the aid of Fig. $4-17$ it can be derived in a simplified form by consid$-$

ering the distance traversed by the rocket $($called the range$)$ to be the integrated area

underneath the velocity$-$time curve. Simplifications here include the assumptions of

no drag, no gravity effect, horizontal flight, relatively small distances traversed dur$-$

ing powered flight compared to total range, and linear increases in velocity during

powered flight:

$t_t=\frac{S+\frac{1}{2}{u}_p{t}_p}{u_0+u_p}$

Here, $S$ is the flight vehicle's range to target corresponding to the integrated area

under the velocity$-$time curve, and $u_p$ is the velocity increase of the rocket during

to target. Best results $($e$.$g$.,$ best hit probability$)$ are usually achieved when the time

to target is as small as practically possible.

Any analysis of missile and propulsion configuration that gives the minimum time

to target over all likely flight scenarios can be complicated. The following rocket

propulsion features and parameters will help to reduce the time to target but their

effectiveness will depend on the specific mission, range, guidance and control system,

thrust profile, and the particular flight conditions.

High initial thrust or high initial acceleration for the missile to quickly reach

a high$-$initial$-$powered flight velocity. See Fig. $12-19.$

Application of a subsequent lower thrust to counteract drag and gravity losses

and thus maintain the high flight velocity. This can be done with a single rocket

propulsion system that gives a short high initial thrust followed by a smaller

$(10$ to $25\%)$ sustaining thrust of longer duration.

For the higher supersonic flight speeds, a two$-$stage missile can be more effec$-$

tive. Here, the first stage is dropped off after its propellant has been consumed,

thus reducing the inert mass of the next stage and improving its mass ratio and

thus its flight velocity increase.

If the target is highly maneuverable and if the closing velocity between missile

and target is large, it may be necessary not only to provide an axial thrust but

also to apply large side forces or side accelerations to a defensive missile. This

can be accomplished either by aerodynamic forces $($lifting surfaces or flying at

an angle of attack$)$ or by multiple$-$nozzle propulsion systems with variable or

pulsing thrusts; the rocket engine then would have an axial thruster and one or

more side thrusters. The side thrusters have to be so located that all the thrust

forces are essentially directed through the center of gravity of the vehicle in

order to minimize turning moments. Thrusters that provide the side accelera$-$

tions have also been called divert thrusters, since they divert the vehicle in a

direction normal to the axis of the vehicle.

Drag losses can be reduced when the missile has a large $\frac{L}{D}$ ratio $($or a small

cross$-$sectional area$)$ and when the propellant density is high, allowing a

smaller missile volume. Drag forces are highest when missiles travel at low

altitudes and high speeds. Long and thin propulsion system configurations and

high$-$density propellants help to reduce drag.

One unique military application is the rocket$-$assisted gun$-$launched projectile for

attaining longer artillery ranges. Their small rocket motors located at the bottom of

gun projectiles must withstand the very high accelerations in the gun barrel $(5000$ to

$10,000g_0\text{'}$s are typical$).$ These have been in production. Derive Eq$.437$; state all your assumptions.