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Local speed of sound as a function of altitude. 

Local speed of sound as a function of altitude. 

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Geography makes ballistic missile flight times between India and Pakistan very short. This has serious consequences for the feasibility and utility of possible early warning systems that could be set up in India or Pakistan to detect incoming missiles from the other side. In this article, we show how one can estimate the time taken for a missile fl...

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... not exactly known, approximate values for most quantities of interest are available from the public record. The one quantity that is usually not speci fi ed is η , the angle between thrust and velocity as a function of time. 79 In solving these equations, we will vary η to obtain the desired range or minimize the time of fl ight. The drag coef fi cient C d during boost phase is assumed to be the same as the measured values for the V-2 rocket and is shown in Figure 3 as a function of the ratio of the velocity of the missile to the local sound velocity. 80 We will assume that the missile jettisons the propellant tanks at the end of the boost phase. In that case all that would be left would be the missile nose cone, which is assumed to be a hemisphere mounted on a blunted cone. The drag coef fi cient C d of the missile during reentry then is given by: C d = 2 sin 2 ( α ) + ( r r n b ) 2 (1 − 2 sin 2 ( α ) − sin 4 ( α )), where α is half the cone opening angle, r n is the radius of the frontal hemisphere and r b is the base radius of nose cone. 81 The density of air and the local speed of sound (needed to calculate the drag coef fi cient) as functions of altitude are taken from the 1976 U.S. standard atmosphere and are shown in Figure 4 and Figure 5. 82 Consider a simple example in which we have, in the earth- fi xed frame, a launch site r B and a target r T , both at the ground level, at the same latitude L and separated by a longitude difference of ψ 0 . As the earth rotates, r B ( t ) and r T ( t ) will both change with time in any fi xed inertial frame. Let us choose the inertial frame as one whose spherical polar angles coincide with the geographical latitude and longitude at t = 0. Then at the start of the missile trajectory, the azimuthal angle separation ψ t = 0 between launch and target is just the longitude difference. As the missile travels the target moves along the azimuthal direction due to earth ’ s rotation, reaching a fi nal azimuthal angle separation ψ given ...

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