How Much Gas Does It Take to Make a Tesla Roadster Hover for 5 Seconds?

A car hovering on gas thrust sounds like a question about engine power. This case shows why it is also a question about carrying—and continuously losing—mass. It follows a hypothetical Roadster-like vehicle through a one-metre ascent, a five-second hover and a controlled descent, checking both the gas budget and the thrust required at every stage.

In the adopted model, 200 kg of gas is enough for the ascent but only 3.370 seconds of subsequent hover. Completing the prescribed two-second ascent, five-second hover and two-second descent requires approximately 347.634 kg of initial gas at the zero-reserve boundary, with dry mass fixed at 1,800 kg. This is a calculation for an idealized propulsion system, rather than a measurement or specification of a Tesla vehicle.

The question: lifting off is only the first requirement

The study separates three nested tasks. Each task includes the preceding stages of the same flight; the vehicle does not receive a fresh tank between stages.

Task	What counts as success	Why the distinction matters
Rise one metre	Reach 1 m after 2 s, with height error at most 0.02 m and vertical-speed magnitude at most 0.02 m/s	Having enough thrust to leave the ground does not establish enough fuel for the remaining mission.
Hover for five seconds	Complete the ascent, then remain within both tolerances for five continuous seconds	The hover consumes gas while vehicle weight steadily falls.
Descend and land	Complete the hover, descend under control, contact the ground at no more than 0.2 m/s and remain grounded	A mathematically smooth touchdown does not establish rebound-free ground contact.

The model also requires nonnegative gas and upward thrust between zero and 35 kN. Once gas is exhausted, available thrust is zero. Continuing a mathematical mass curve below the retained vehicle mass would not represent a possible flight.

What the model assumes

Input	Baseline	Added-payload comparison
Dry vehicle mass	1,800 kg	1,800 kg
Retained payload	0 kg	200 kg
Initial gas mass	200 kg	200 kg
Initial total mass	2,000 kg	2,200 kg
Maximum upward thrust	35 kN	35 kN
Effective exhaust velocity	500 m/s	500 m/s
Gravity	9.81 m/s²	9.81 m/s²
Target height	1 m	1 m
Planned ascent / hover / descent	2 / 5 / 2 s	2 / 5 / 2 s

Motion is vertical only. Exhaust velocity is held constant, thrust responds instantly, and the model omits wind, drag, tilt, tank-pressure changes and actuator delay. The gas-load scan keeps the 1,800 kg dry mass fixed even when more gas is added: larger tanks and their installation mass are not included.

How the calculation works

The vehicle follows a prescribed smooth path, rather than a simulated flight controller. The quintic shape q(s) = 10s³ − 15s⁴ + 6s⁵ produces an ascent that starts and ends at rest, a constant-height hover, and a corresponding descent. Velocity and acceleration vanish at the phase boundaries.

For instantaneous total mass m, acceleration a, upward thrust T and effective exhaust velocity vₑ, the governing relations are:

Required thrust:       T = m × (g + a)
Gas-driven mass loss:  dm/dt = −T / vₑ
Mass along the path:  m(t) = m₀ × exp(−[g × t + v(t) − v(0)] / vₑ)
Gas remaining:        m(t) − retained vehicle mass

Hovering therefore does not require constant thrust: gravity stays constant, but weight decreases as gas is expelled. For a rest-to-rest trajectory lasting t_f, the ideal minimum gas budget is retained mass × [exp(g × t_f / vₑ) − 1]. The full target flight lasts nine seconds. This endpoint budget must still be checked against the thrust limit throughout the trajectory.

The saved study first used fourth-order Runge–Kutta integration and then compared it with the analytic trajectory and mass law. Supplemental scans refine gas-exhaustion and thrust-limit crossings. These checks test numerical consistency; exact tracking of a prescribed path does not demonstrate a working controller.

What happens with 200 kg of gas

The baseline begins with 19.620 kN of weight and a maximum thrust-to-weight ratio of 1.784. It can complete the prescribed ascent without using the full 35 kN capacity.

Stage or requirement	Recorded result	Interpretation
Two-second ascent	Reaches 1 m with effectively zero vertical speed	The first task is feasible in the ideal model.
Ascent gas consumption	76.960 kg	123.040 kg remains for hover and descent.
Fueled hover after ascent	3.370 s	Falls short of the requested five seconds.
Exhaustion time from launch	5.370 s	Occurs before the planned 7 s hover endpoint.
Controlled descent	Not reached	A zero in an uncompleted descent budget does not mean descent needs no gas.
Ground-contact behavior	Not modeled	No conclusion about rebound or structural safety follows.

If thrust is set to zero after exhaustion, an ideal fall from one metre at rest takes 0.452 s and reaches 4.429 m/s at first contact. That is far above the study's 0.2 m/s contact-speed limit. This separate ballistic calculation is a failed uncontrolled fall, not completion of the planned descent.

Does carrying more gas solve the problem?

More gas extends endurance, but it also increases the mass that must be lifted. The saved gas-load scan captures both effects for the same prescribed nine-second flight.

Initial gas	Required peak thrust	Recorded outcome
100 kg	21.192 kN	Gas exhausted 2.756 s after launch.
200 kg	22.307 kN	Gas exhausted at 5.370 s.
300 kg	23.422 kN	Gas exhausted during descent, at about 7.952 s.
347.634 kg	23.954 kN	Ideal zero-reserve touchdown.
400 kg	24.538 kN	Ideal touchdown with 43.89 kg gas remaining.
1,000 kg	31.230 kN	Ideal touchdown with 546.77 kg remaining.
1,300 kg	34.576 kN	Prescribed airborne trajectory remains within the thrust cap.
1,400 kg	35.691 kN	Required thrust exceeds the cap about 0.232 s into ascent.
1,800 kg	40.153 kN	Initial weight alone exceeds 35 kN, preventing immediate lift-off.

The resulting trajectory-feasible interval is approximately 347.634–1,338.010 kg of gas. Its lower end comes from fuel consumption; its upper end comes from peak thrust demand. This interval applies to this trajectory with unchanged dry mass. It is not an optimized design range, and the lower boundary contains no operational reserve.

Shorter hover or extra payload: two different changes

With the original 200 kg gas load, shortening hover to one second permits the prescribed descent and leaves about 13.12 kg of gas. The baseline's maximum fueled hover is 3.370 s if no descent reserve is held back, but only 1.370 s when the two-second descent is included.

Adding 200 kg of retained payload moves the budget in the opposite direction:

Quantity	Baseline	With 200 kg payload
Ascent gas consumed	76.960 kg	84.656 kg
Gas left after ascent	123.040 kg	115.344 kg
Fueled hover without descent reserve	3.370 s	2.858 s
Hover with prescribed descent reserve	1.370 s	0.858 s
Minimum gas for the full nine-second path	347.634 kg	386.260 kg

The payload version still completes the ascent, but its five-second hover fails earlier. Retained payload matters throughout the flight, while gas mass gradually disappears.

What is preserved in the research record

Evidence	What it supports
final_research_report.md	Consolidated assumptions, methods, results and corrections.
baseline_phase_budgets.csv and baseline_task_timeseries.csv	Shared ascent, partial hover, exhaustion and phase-by-phase gas accounting.
final_gas_scan.csv and final_failure_cases.csv	Gas-load outcomes and the first constraint reached in failure variants.
hover_budgets_v2.csv	Comparisons of one-, five- and ten-second requested hovers.
payload200_parameters.json, budgets and time series	Explicit retained-payload version.
Saved analysis scripts, PNGs and PDFs	Numerical checks and graphical presentation of the recorded calculations.

The original study records analytic checks for zero thrust, hover, constant acceleration and constant thrust, alongside timestep refinement. For example, ascent height discrepancies were approximately 1.64 × 10⁻¹¹ m at a 4 ms step and 1.04 × 10⁻¹² m at a 2 ms step. Such small numerical discrepancies do not quantify uncertainty in real propulsion hardware.

What the case establishes

The useful result is a constrained budget: lifting off, hovering and returning under control are separate requirements, and added gas helps only while its weight remains compatible with available thrust. The baseline completes the first requirement and fails the five-second hover; a larger gas load can complete the prescribed airborne path in the ideal model.

Measured blowdown and mass-flow curves, credible tank and installation masses, disturbance control, and a contact model are still needed to evaluate an actual vehicle. Neither a zero-reserve touchdown nor numerical agreement establishes a practical flying Roadster.