3 ms·
I'll take a stab. The lift-to-drag ratio decreases with speed, but the limit is not zero, but rather 4. https://en.wikipedia.org/wiki/Lift-to-drag_ratio#Super
by credit_guy 4y ago
I'll take a stab.
The lift-to-drag ratio decreases with speed, but the limit is not zero, but rather 4.
https://en.wikipedia.org/wiki/Lift-to-drag_ratio#Supersonic https://en.wikipedia.org/wiki/Lift-to-drag_ratio#Supersonic
More precisely the L/D ratio as a function of the Mach number M is (4M+3)/M. Of course, that's the ideal maximum L/D ratio. But let's go with it.
So, let's see what happens if you travel at Mach 2, Mach 5 (lower limit of the hypersonic regime) and Mach 10 (which is around the reported speed of the Russian Zircon missile, which they tested a few days ago, on 28May2022). The L/D ratios would be 5.5, 4.6 and 4.3.
Let's say we envision a flight path of 1000 km (same as the said Zircon test). At Mach 10 that would take about 300 seconds = 5 min, and at the slower speeds of Mach 5 and Mach 2 it would take 10 and 25 minutes.
Of course, you'd like the missile to get to the target sooner rather than later, if you could help it. The paradoxical thing is that the faster the missile is, the less fuel it needs to burn.
You read that right. And this has nothing to do with air-breathing scarmjet engines. It's just old fashion rocket engines.
Here's how this works, in our example. At Mach 2, our drag is 1/5.5 = 18.2% of the lift, so your engines need to push with 0.18g to keep you at constant speed. They need to do that for 1500 seconds, so that translates into a delta-v of 0.18 x 10 x 1500 = 2.7 km/s. A typical kerosene-oxygen rocket engine has an exhaust velocity around the same, so the rocket equation tells us the ratio of dry mass over total mass (dry mass + fuel) is about exp(-1)= 37%. So, 63% of your missile has to be fuel.
At Mach 5 the drag is 1/4.6 = 0.22g. Delta-v becomes 0.22 x 10 x 600 = 1.3 km/s, and the ratio of dry mass to total mass completely reverses: now only 62% of the vehicle mass is dry, and 38% is fuel.
At Mach 10, delta-v is 0.23 x 10 x 300 = 0.7 km/s and the fuel mass is only 23%.
For comparison, the phenomenal Tomahawk cruise missile has a fuel weight of about 450 kg and a total mass of 1300 kg, so a fuel ratio of about 35%. A Mach 10 hypersonic missile could have a comparable fuel ratio as a Tomahawk subsonic cruise missile.
Of course, the Mach 10 missile needs a booster to bring it at altitude and cruising speed. But that booster can be a fairly dumb (and inexpensive) rocket. And in the era of SpaceX, who knows, that booster might even be reusable.