The Manual Burn

Convert everything to one consistent unit system before you multiply anything — SI is easiest. Don’t plug pounds-force and feet-per-second straight into the formula; the mixed units are the trap here, not the algebra.

Newton’s second law, rearranged: force equals mass times acceleration, and acceleration is just velocity change over time. So:

F = m · (Δv / t)   ->   t = m · Δv / F

Unit conversions needed to get everything into SI before using it:

1 lbf = 4.4482216153 N
1 ft  = 0.3048 m

Answer: ≈ 22.4 seconds

That’s longer than the roughly 14 seconds Mission Control actually commanded — not an error in the arithmetic, but a limit of the model. The real burn had three phases (a short RCS thruster firing to settle the propellant, the main descent-engine burn, then a further RCS trim), and the ship was already lighter than its launch mass, having burned propellant on two earlier course corrections. A single clean formula can’t reproduce all of that; it gets you in the right neighborhood, which is exactly the kind of estimate a slide rule was built for.

43975  ENTER
7.8    ×             m × Δv(ft/s) = 343005
0.3048 ×             × ft→m = 104547.924  (m · Δv, in SI)
1050   ENTER
4.4482216153  ×       F in newtons = 4670.6327
÷

Display: 22.384103

Same formula. One habit worth having on this machine: always type a leading 0 before a decimal fraction that starts a fresh number, e.g. 0.3048 — starting directly with the point on its own doesn’t register.

43975  ENTER
7.8    ×
0.3048 ×
1050   ENTER
4.4482216153  ×
/

Display: 22.384102

Identical RPN structure to the 42S.

43975  ENTER
7.8  ×
.3048  ×
1050  ENTER
4.4482216153  ×
÷

Result: ≈ 22.3841 → 22.4 s.

43975  ENTER
7.8    ×
.3048  ×
1050   ENTER
4.4482216153  ×
/

Display: ≈ 22.3841 → 22.4 s.

This is the challenge where RPL is genuinely a different tool rather than RPN with extra menus. The HP-48 carries units on the stack as part of the object: a number and a unit expression joined by an underscore, like 2_in or 8.303_gal/h. You don’t type the underscore — put the number in the command line, then pick the unit off the UNITS catalog (→ UNITS), which attaches it.

So the two conversion constants this challenge is built around — 0.3048 for ft→m and 4.4482216153 for lbf→N — never get looked up or typed at all. Tag the raw figures with the units they were actually recorded in, do the arithmetic, and finish with UBASE (← UNITS), which reduces whatever compound unit you’ve ended up with to SI base units:

43975                      → UNITS  MASS   KG      43975_kg
7.8                        → UNITS  SPEED  FT/S    7.8_ft/s
×                                                  343005_kg*ft/s
1050                       → UNITS  NXT  FORCE  LBF  1050_lbf
÷                                                  326.671428571_kg*ft/(s*lbf)
UBASE                      ← UNITS  UBASE          22.3841031406_s

The mixed kg*ft/(s*lbf) is already the right answer in the wrong clothes — dimensionally it is a time, it just isn’t in seconds yet. UBASE is what notices that and collapses it, unit and all, to 22.3841031406_s → 22.4 s, matching the HP-42S value above.

(UVAL, two keys along the same menu, strips the unit tag off and leaves the bare number, if you want it without the _s.)

Algebraic entry, one line:

(43975*7.8*.3048)/(1050*4.4482216153)

Result: ≈ 22.3841 → 22.4 s.

Same single expression as the TI-84, natural display mode:

(43975×7.8×0.3048)÷(1050×4.4482216153)

Result: ≈ 22.3841 → 22.4 s.

Only multiplication and division here — no log or trig scales needed, just the C and D scales (and the CI scale saves a step, if your rule has one).

1. Convert first, since a slide rule can’t do unit conversion for you: Δv = 7.8 ft/s ≈ 2.38 m/s, F = 1050 lbf ≈ 4670 N. (Do these conversions by simple multiplication on the rule, or by hand — either is fine, they’re each one multiply.)
2. Set the cursor on 4.40 (representing 43,975, tracking the exponent separately) on the D scale, multiply by 2.38 using the C scale: read off ≈ 1.05, exponent works out to ×10⁵ — that’s m·Δv.
3. Without moving the cursor, slide the C scale to divide by 4.67 (representing 4670): read ≈ 2.24 on D, exponent ×10¹.
4. Result: ≈ 22.4 s — as much precision as three digits ever gives you on a rule, which is exactly why the answer is stated to 3 significant figures rather than a longer decimal.