Roughly how fast would Falcon 9 accelerate, and what would its payload's final speed be, if mounted horizontally on an ideal cart? Make simplifying assumptions as needed: massless frictionless cart, engines and both stages optimized for atmosphere rather than vacuum (or travelling in an evacuated Hyperloop), etc.

Use one cart per stage if you like (it doesn't matter, if carts are massless.)

This is just to compare it to terrestrial vehicles, of which there are more examples than vehicles which climb vertically. Horizontal vs. horizontal performance is fairer than what is considered by a similar question.

  • $\begingroup$ Cool question, but needs one detail; first stage both? I'm trying to figure out how many carts to buy. :-) $\endgroup$ – uhoh Nov 23 '20 at 22:01
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    $\begingroup$ What's the payload mass? How precise an answer are you looking for? $\endgroup$ – Russell Borogove Nov 23 '20 at 23:10
  • $\begingroup$ At the moment of lift-off the acceleration is just barely greater than 1g. Fast cars do better than that. $\endgroup$ – hdhondt Nov 23 '20 at 23:26
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    $\begingroup$ @RussellBorogove Well, 1.2 g is just barely greater than 1 g. $\endgroup$ – hdhondt Nov 24 '20 at 2:08
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    $\begingroup$ The biggest difficulty all "land speed record" vehicles have is staying in contact with the ground. Escape velocity is 11.2 km/s , so when you get near that speed, you'll "go to space today" regardless. $\endgroup$ – Carl Witthoft Nov 24 '20 at 12:29

Using SpaceX numbers and Space Launch Report estimates for those missing, here are the key parameters for the active Falcon 9 Block 5:

  • Merlin atmosphere optimised sea level ISP: 283s
  • Merlin atmosphere optimised sea level thrust: 845kN
  • Merlin vacuum optimised ISP: 348s
  • Merlin vacuum optimised ISP thrust: 934kN
  • Stage 1 dry mass: 27.2 tonnes
  • Stage 1 total mass: 445.9 tonnes
  • Stage 2 dry mass: 4.5 tonnes
  • Stage 2 total mass: 116.0 tonnes

For a payload, since the purpose is to compare to terrestrial vehicles, I'm assuming a typical car of 1.5 tonnes.

Since this is a "spherical cow" question, I'm also ignoring drag, which would otherwise be somewhat notable while travelling at multiple km/s at sea level.

From these numbers, the acceleration at 4 relevant points for both configurations are directly available (assuming no throttling):

  • Stage 1 ignition SL|Vac: 13.5m/s² | 14.9m/s²
  • Stage 1 burnout SL|Vac: 52.6m/s² | 58.1m/s²
  • Stage 2 ignition SL|Vac: 6.45m/s² | 7.13m/s²
  • Stage 2 burnout SL|Vac: 140.8m/s² | 155.7m/s² (throttling looks like a good idea here though...)

Since the Falcon doesn't stay at constant atmospheric pressure for the whole flight, for the purpose of the question the burn times have to be re-calculated:

$$T_{burn} = \frac{m_{propellant}\cdot I_{SP}\cdot g_0}{F_{thrust}}$$

Which for the relevant configurations yield:

  • Stage 1 sea level: 153s
  • Stage 1 vacuum: 170s
  • Stage 2 sea level: 366s
  • Stage 2 vacuum: 408s

Furthermore, the total velocity change for each stage is required, as given by the rocket equation:

$$\Delta v = \ln\left(\frac{m_1}{m_0}\right)\cdot I_{SP}\cdot g_0$$

($m_0$ is payload + stage 1 dry mass + stage 2 mass for the first stage, payload + stage 2 dry mass for the second stage)

  • Stage 1 sea level: 3,770m/s
  • Stage 1 vacuum: 4,640m/s
  • Stage 2 sea level: 8,260m/s
  • Stage 2 vacuum: 10,160m/s
  • Sea level total: 12,030m/s
  • Vacuum total: 14,800m/s

Then we have enough values to get average acceleration:

  • Stage 1 sea level average: 24.6m/s²
  • Stage 1 vacuum average: 27.3m/s²
  • Stage 2 sea level average: 22.6m/s²
  • Stage 2 vacuum average: 24.9m/s²
  • Full rocket sea level average: 23.2m/s²
  • Full rocket vacuum average: 25.6m/s²
  • $\begingroup$ I'd probably stop at or near Max-Q , but I suppose for the purpose of a theoretical study we'll assume an indestructible rocket body. $\endgroup$ – Carl Witthoft Nov 24 '20 at 12:26
  • $\begingroup$ So, escape velocity after all. (Sideways or upwards doesn't matter.) $\endgroup$ – Camille Goudeseune Nov 24 '20 at 16:58

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