Bloodhound Going Like a Bat Out of Hell
Calculating the Distance for the Missile to Become Supersonic

Pete Murray
Various web pages claim that the Bloodhound missile was supersonic only a few metres from the launcher. However, the physics appears to say otherwise.

Calculating the time to become supersonic.
The speed of sound at sea level is approximately 340 metres per second.
The missile has a mass of approximately 5000lb and the 4 boost motors deliver approximately 100,000lb of thrust, so the initial acceleration is 20g, or 196m/s2.

If we simplify things a bit and assume that the acceleration remains constant, we can calculate that the missile becomes supersonic after 1.73 seconds.

Calculating the distance travelled.
Plugging the numbers into Mr Newton’s equation, d = ½ at2, gives a distance of 293 metres.

I’ve taken a few liberties here with assumptions and simplifications, but 296 metres is hopefully not too far off the mark. Does anyone know the actual numbers?

As a matter of interest.
The missile gyros were all subjected to a series of drop tests (20g I think) before final testing. Radio Frequency Units (RFU) were also drop tested and monitored to ensure that the local oscillators and IF stages remained stable during the deceleration.

The LCP Doppler prediction calculation coded into the LCP software assumes a missile velocity of 2400 feet per second at acquisition of the target (5 seconds after launch). This converts to 731 m/s or 1636 mph.

Comments Received:
From Richard Vernon:
The Mk 2 missile was only stressed to 35g in the longitudinal axis so Zero to Mach one in its own length (around 1700g) was a bit of Newton exaggeration. Actual acceleration at launch was 19 to 21g dependent on the ambient temperature of the boost rocket motor propellant charge.

Actual specs for the Gosling XV motor were 23,000 lbs for 3.8 seconds at -25°C to 31,500 lbs for 2.8 seconds at +40°C. The only firing that I do have information about in the UK was the Singapore round fired in 1980. It was fired at a sea level temperature of 12.6°C and an atmosphere pressure of 1007.3 millibars. Boost motor separation was at 3.803 seconds after first movement at which point the missile was 1332.6 m (4536.1 ft) from the launcher in ground distance and 795.0 m (2608.3 ft) above the launcher at a speed of 701.0 m/sec (2299.9 fps) (1568.09 mph, 1362.63 Knots, Mach 2.04373). At that point the missile angle to the ground was 29.060° and it had rolled to starboard 15° off the line of fire (which was 328°), which wasn't a surprise seeing that the wind velocity was 5.182 m/s (11.59 mph) from a bearing of 250° (almost side on to the line of fire). For hotter (and colder) temperatures, the Swedes fired one at -25° C from Vidsel, while all of the hotter ones would have been from Woomera.

The problems getting figures from those firings is most of them were done with the original Gosling IV motor planned for BH2, which only produced 22,000 lbs at -25°C and because of that was not capable of accelerating the missile to a speed where the ramjets could produce enough thrust to accelerate the missile post-boost separation on a really cold day (which was bad news for sales to the Swedes and the Swiss).      
         

Bloodhound System