Developing intuition about altitude and velocity in multi stage rockets

Question

I just started learning some basics of rocketry and am struggling to understand the outcomes of a simulator I’m programming. All other things being equal, when I attempt to simulate the altitude over time of a single stage rocket with the below attributes, its peak altitude is only a tiny bit less (~97.5%) than if that rocket were segmented into two stages (also below). Note, both instances have the same total dry mass and fuel mass.

Multi stage :

• First stage : ISP = 282s, dry mass = 22226kg, diameter = 3.66m, constant thrust = 7600000N, fuel mass = 522629kg, thrust angle = 5 degrees
• Second stage : ISP = 348s, dry mass = 4000kg, diameter = 1.7m, constant thrust = 934000N, fuel mass = 58069kg, thrust angle = 5 degrees

Single stage:

• First stage : ISP = 282s, dry mass = 26226kg, diameter = 3.66m, constant thrust = 7600000N, fuel mass = 580698kg, thrust angle = 5 degrees

Is this a reasonable outcome or is my code possibly bugged? I was under the impression multi staging led to significantly greater delta V, and thus max altitude (in this simulation's case) gains. Also, if the entirety of the second stage’s mass were treated as just additional dry mass for the single stage instance, the peak altitude it achieves is nearly identical to the peak altitude of the first stage in the two stage instance. This seems intuitive. But the tiny total performance gain of multi staging over single staging doesn’t for me. If the simulation is wrong, what sort of gains should be expected in a case like this?

Answer 1

I will argue that your simulation outcome is not reasonable.

Consider what happens if we decouple your multi stage rocket right at launch, making no use of the first stage at all. That is, just using the second stage as a single stage rocket.

Its delta-v:

$$\Delta v = I_{sp} \cdot g \cdot \ln{\left(\frac{M_1}{M_0}\right)} = 348s \cdot 9.81m/s^2 \cdot \ln{\left(\frac{58069kg + 4000kg}{4000kg}\right)} = 9360m/s$$

With a thrust to weight ratio at launch of 1.54

The single stage rocket:

$$\Delta v = 282s \cdot 9.81m/s^2 \cdot \ln{\left(\frac{580698kg + 26226kg}{26226kg}\right)} = 8690m/s$$

With a thrust to weight ratio at launch of 1.28

Thus, the second stage alone should be a more capable rocket than your single stage rocket, before even considering the first stage. There must therefore be something up with your simulation.

Answer 2

I agree with the answer of SE that your results appear to be incorrect.

I suggest that you break your simulation down into discrete testable steps as follows:

1. Implement thrust only without any angle

   • Consider a single stage rocket

   • Verify that the velocity of the rocket at burnout in your simulation is equal to that calculated by the Tsiolkovsky (rocket) equation

   • Now consider a multi-stage rocket and verify with the multi-stage Tsiolkovsky equation

2. Implement gravity only

   • Consider a projectile launched with a certain initial velocity

   • Verify with projectile equations (HyperPhysics is a good source)

   • You can move from 1D motion to 2 or 3 dimensions now and verify

3. Implement thrust with gravity without any angle

   • Consider a single stage rocket

   • Verify that the velocity of the rocket at burnout in your simulation is equal to that calculated by the Tsiolkovsky equation minus $t\,g$ where $t$ is the time since launch and $g$ is $9.81\,\mathrm{m/s^2}$.

4. Further additions such as thrust angle and aerodynamic forces will make it harder and harder to verify with analytical equations. Therefore, I suggest you implement these additions one-by-one and compare with the previous results to check whether the change is reasonable or not.