Procrastination rewarded

A black powder booster motor, one ending in “-0”, has no delay charge, no ejection charge, no clay cap. When the propellant burns through it sends hot combustion products forward where they can ignite the next stage’s motor.

What if you used a non booster motor in a booster stage? What would happen? Bad things, probably. The ejection charge and clay cap combination don’t make for a good way to ignite another motor. And then there’s the delay…

Is the delay bad?

Well, that depends. Short answer: It might not be. Again, the ejection charge and clay cap are bad, so don’t do it; but what about delaying the staging?

My understanding is that competition flyers use delayed staging. (Custom made motors, apparently.) Let’s see if we can figure out why.

Let’s suppose we drag race two otherwise identical two-stage rockets; on one, staging occurs immediately after booster burnout, while on the other, staging is delayed. The boosts are the same, so both have the same altitude and same velocity at the end of boost. Now Rocket 1 fires its sustainer and continues accelerating; once the sustainer burns out, Rocket 1 coasts. Meanwhile Rocket 2 coasts, then fires its sustainer and accelerates. At the end of Rocket 2’s sustainer burn, both rockets have had the same boosts and both have coasted the same amount of time, just in opposite order. They end up having the same velocity at that moment… if we neglect air drag. Which we will, for the moment.

So now they’re both flying at the same speed, both decelerating due to gravity at the same rate, so both reach apogee at the same time and fall back down. But between booster burnout and Rocket 2 sustainer burnout, Rocket 1 accelerated to its maximum speed and then decelerated, while Rocket 2 reached the same speed by first decelerating, then boosting to its maximum speed. During that whole time, Rocket 1 was flying faster than Rocket 2, and so it got a head start. After Rocket 2 sustainer burnout it’ll keep that head start all the way to apogee and back down. Rocket 1 flies higher.

Here’s a graph:image (3)

Both boosters fire the same amount at the same time (dark blue line), then Rocket 1 sustainer fires (dark red), then Rocket 2 sustainer (purple). The velocities are shown in gold (Rocket 1) and light blue (Rocket 2). They’re the same (so all you see is light blue) from launch to booster burnout, and from Rocket 2 sustainer burnout to end of flight, but in between Rocket 1 has higher velocity than rocket 2.

So the altitudes are the same through booster burnout, but then Rocket 1’s height (green) increases faster than Rocket 2’s (pink) and Rocket 1 stays above Rocket 2 from there on.

(This graph isn’t intended to be a realistic simulation of any actual rocket, by the way. It’s calculated using correct equations but completely made up numerical values. It’s just to illustrate what happens.)

So delayed staging seems to be a losing proposition. But wait, I said we were neglecting air drag. What happens if we don’t?

Air drag depends on velocity. So when Rocket 1 starts going faster than Rocket 2, it starts experiencing higher drag. Both have the same sustainer thrust and the same gravitational acceleration, but because Rocket 1 experiences more drag, they do not end up with the same velocity at Rocket 2 sustainer burnout. In fact Rocket 1’s velocity matches Rocket 2’s earlier, and continues to decrease, while Rocket 2, still under sustainer boost, continues to go faster. At Rocket 2 sustainer burnout, Rocket 2 is going faster!

See the next graph:image (2)

 

 

Rocket 1 velocity (gold) goes higher than Rocket 2 (light blue), but doesn’t increase as much under sustainer thrust and drops more under coast. The two velocities cross during Rocket 2 sustainer thrust, and after that, Rocket 2 is going faster.

Once Rocket 2’s sustainer burns out they’re both being decelerated by gravity and by drag. Rocket 2 now being faster, its drag deceleration is higher, so it slows down faster. It never gets slower than Rocket 1, though. (If it did slow down to Rocket 1’s speed, its drag deceleration would become equal, and both would keep the same velocity from there on. But even that doesn’t happen. They both approach terminal velocity asymptotically, with Rocket 2 continuing to fly faster than Rocket 1 albeit by a smaller and smaller amount.)

So how about the altitude? During the sustainer boosts, Rocket 1 (green) gets higher than Rocket 2 (pink). At Rocket 2 sustainer burnout, Rocket 1 has the lead, but Rocket 2 is moving faster. Can it catch up?

In this graph, it does. Rocket 2’s higher post-burnout velocity takes it past Rocket 1 and up to a slightly higher apogee.

Now, this second graph is quantitatively even less realistic than the first. For one thing, I neglected the fact that drag deceleration depends on rocket mass, and Rocket 1 loses mass sooner than Rocket 2. For another thing, I used drag proportional to velocity squared, which is correct at high velocity (e.g. under thrust), but not at low velocity (e.g. at apogee). But the qualitative features here are correct.

There’s obviously a balancing act going on here. Delay staging too long, and you fall too far behind to catch up. Delay it too little, and you don’t lose enough velocity to get enough reduction in drag. For any given rocket with any given thrust curves under any given conditions, there’s some optimum staging delay (which could be zero!) giving the highest apogee.

There are additional risks, too. Delay staging too long and the velocity could fall below the point where the fins can maintain aerodynamic stability. The rocket can start tumbling, and then light the sustainer while pointing sideways. Or down! Even if the rocket isn’t tumbling, you’re giving it that much longer to weathercock before sustainer ignition, which could have a similar effect. Combine that with the fact that it requires custom rocket motors (or plugged motors and electronic ignition; or motors whose caps and ejection charges have been removed, which violates the rocketry safety code) and that the apogee increase is usually fairly small (not anything like factors of two), and it’s clear delayed staging is pretty much for competition experts only. Me, I’m content to just draw the graphs.

 

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