RPN: HP 11C: Surface Gravity and Escape Velocity

RPN: HP 11C: Surface Gravity and Escape Velocity

EQUATIONS

The surface gravity constant of a celestial object (planet, dwarf planet, star, etc.):

g_p = G * M ÷ R²

The escape velocity of a celestial object:

v_esc = √(2 * G * M ÷ R)

where (using SI units):

g_p: surface gravity (m/s)

v_esc: escape velocity (m/s)

M: (measured) mass of the object (kg)

R: (average) radius of the object (m)

G: Universal Gravitational Constant (G ≈ 6.6743 * 10^-11 N m²/kg² (or m³/(s² kg))

The value of G is the 2022 CODATA value (https://physics.nist.gov/cgi-bin/cuu/Value?bg)

Determining Surface Gravity and Escape Velocity

DERIVATION - Determine the surface gravity constant in terms of escape velocity.

Start with the escape velocity:

v_esc = √(2 * G * M ÷ R)

(v_esc)² = 2 * G * M ÷ R

dividing both sides by 2 (we'll see why this important in a bit):

(v_esc)² ÷ 2 = G * M ÷ R

(v_esc)² * 1/2 = G * M * 1/R

Then insert the square of escape velocity in the equation for the surface velocity:

g_p = G * M ÷ R²

g_p = G * M * 1/R²

g_p = G * M * 1/R * 1/R

g_p = (v_esc)² * 1/2 * 1/R

g_p = (v_esc)² ÷ (2 * R)

The equations will the be:

v_esc = √(2 * G * M ÷ R)

g_p = (v_esc)² ÷ (2 * R)

Set the stack up as:

Y: M (mass, kg)

X: R (radius, m)

The results are shown in the stack:

Y: g_p (surface gravity, m/s²)

X: v_esc (escape velocity, m/s)

Algorithm (done with an HP 11C):

ENTER

ENTER

R↑

2

×

6.6743e-11 (Keys: 6 . 6 7 4 3 EEX 1 1 CHS)

×

R↑

÷

ENTER

√ (view escape velocity)

R↓

x<>y

÷

2

÷ (view surface gravity)

R↑ (set surface gravity in the Y stack, escape velocity in the X stack)

Example:

Estimate the surface gravity constant and escape velocity of Venus.

Venus

Mass ≈ 4.8675 * 10^24 kg

Radius ≈ 6.0518 * 10^6 m

Surface gravity ≈ 8.8704 m/s

Escape velocity ≈ 10361.6414 m/s

Determining a Planet's Radius and Escape Velocity

Problem: Given Earth's surface gravity is defined as 9.80665 m/s and mass of 5.972168 * 10^24. Estimate the radius and escape velocity.

Here we are given g_p and M, and we are tasked with finding R and v_esc.

Start by solving for R:

g_p = G * M ÷ R²

Multiply by R² and divide by g_p. Keep this in mind.

R² = G * M ÷ g_p

Take the square root and solve for the radius.

R = √(G * M ÷ g_p)

Note that:

R² = G * M ÷ g_p

g_p * R² = G * M

2 * g_p * R² = 2 * G * M

2 * g_p * R = 2 * G * M ÷ R

This makes for an easy substitution for v_esc.

v_esc = √(2 * G * M ÷ R) = √(2 * g_p * R)

The equations used are:

R = √(G * M ÷ g_p)

v_esc = √(2 * g_p * R)

The algorithm uses one memory register, I just picked R0 (done with an HP 11C):

STO 0

÷

6.6743e-11 (Keys: 6 . 6 7 4 3 EEX 1 1 CHS)

×

√ (view R)

ENTER

RCL 0

×

2

×

√ (view v_esc)

Set the stack up as:

Y: M (mass, kg)

X: g_p (surface gravity, m/s²)

The results are shown in the stack:

Y: R (radius, m)

X: v_esc (escape velocity, m/s)

Results:

Inputs:

Mass of Earth ≈ 5.972168 * 10^24 kg (enter as the y stack)

Surface Gravity = 9.80665 m/s² (enter as a x stack, and yes, surface gravity of Earth is defined to be exactly 9.80665 m/s²)

Outputs:

Y: Radius of Earth ≈ 6375416.060 m

X: Escape Velocity ≈ 11182.2604 m/s

Sources

The NIST Reference on Constants, Units, and Uncertainty. "Newtonian constant of gravitation" Fundamental Physical Constants. Last updated May 9, 2024. https://physics.nist.gov/cgi-bin/cuu/Value?bg Retrieved September 4, 2025.

Research & Education Association. The Essentials of Astronomy Piscataway, New Jersey. 2004. ISBN 0-87891-965-1

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

TI-74: Lottery, Solving Linear Equation Script, Escape Velocity Solver

TI-74:  Lottery, Solving Linear Equation Script, Escape Velocity Solver

TI-74 Program:  Lottery

This program collects six random digits from 1 to 63.  The results are stored in an array N.  The dimension pointers of the TI-74 are from 0 to n-1. 

To execute:  RUN 200

Note:  comments (after !) are for notes, and do not need to be typed. 

200 DIM N(5)
202 INPUT "SEED: ";S
204 RANDOMIZE S
210 FOR I=0 TO 5    ! choose random numbers
212 R=INT(RND*63+1)
220 FOR J=0 TO 5   ! check to see if the random number is unique
222 IF R=N(J) THEN 212
224 NEXT J
230 N(I) = R
232 K=K+1
234 PRINT "# "&STR$(K)$": "&STR$(N(I))
236 PAUSE 1.5
240 NEXT I
242 END

TI-74 Program:  Solving Linear Equation Script

This program uses READ and DATA to execute a teaching script.

To execute:  RUN 300

300 FOR I=1 TO 5
302 READ S$  ! string variables have the dollars sign after the name
304 PRINT S$: PAUSE
306 NEXT
310 DATA "To Solve: a*x + b = c"
312 DATA "a*x + b - b = c - b"
314 DATA "a*x = c - b"
316 DATA "(a*x)/a = (c-b)/a"
318 DATA "x = (c-b)/a"
320 END

TI-74 Program:  Escape Velocity Solver

This program is in two stages:
1.  Enter data.  Use M, R, or V to enter data.  When done, press C to solve.
2.  Choose which variable to solve for. 

M = mass of the planet in kg
R = radius of the planet in km
V = escape velocity in km/s

G = Universal Gravitational Constant
= 6.672E-11 m^3/(kg*s^2) = 6.672E-20 km^3/(kg*s^2)

Equation:  V = √(2 * G * M/R)

To execute:  RUN 400

400 M=0:R=0:V=0:G=6.674E-20
402 PRINT "M MASS, R RADIUS, V VEL, CALC":K$=KEY$
406 IF K$="M" THEN 420
408 IF K$="R" THEN 430
410 IF K$="V" THEN 440
412 IF K$ = "C" THEN 450
414 GOTO 402   ! repeats loop if any other key is pressed

420 INPUT "MASS (KG): ";M
422 GOTO 402
430 INPUT "RADIUS (KM): ";R
432 GOTO 402
440 INPUT "VELOCITY (KM/S): ";V
442 GOTO 402
450 PRINT "SOLVER: M,R,V": K$=KEY$

452 IF K$="M" THEN 460
454 IF K$="R" THEN 470
456 IF K$="V" THEN 480
458 GOTO 450

460 M=V^2*R/(2*G)
462 PRINT "MASS =";M;" KG": PAUSE
464 GOTO 490
470 R=2*G*M/V^2
472 PRINT "RADIUS =";R;" KM": PAUSE
474 GOTO 490
480 V=SQR(2*G*M/R)   ! SQR is √
482 PRINT "VELOCITY =";V;" KM/S": PAUSE
484 GOTO 490

490 PRINT "AGAIN? Y/N": K$=KEY$
492 IF K$="Y" THEN 402
494 IF K$="N" THEN END
496 GOTO 490

Examples: 

Solve for V:   M: 5.9724E24 kg, R:  6374 km,  Result: V ≈ 11.178 KM/S
*Escape Velocity for Earth

Solve for M:  R: 5800 km, V: 16 km/s,  Result:  M ≈ 1.112E25 kg

Solve for R:  M: 3.8E24 kg, V: 22 km/s, Result:  R ≈ 1048.0148 km

Eddie

All original content copyright, © 2011-2019.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

TI-55 III Programs Part I: Digital Root, Complex Number Multiplication, Escape Velocity

TI-55 III Programs Part I:  Digital Root, Complex Number Multiplication, Escape Velocity

This blog begins a three part series of programs with the TI-55 III.  Let's show what this calculator can do.

For Part II:  Impedance of a Series RLC Circuit, Quadratic Equation, Error Function

For Part III: Area and Eccentricity of Ellipses, Determinant and Inverse of 2x2 Matrices, Speed of Sound/Principal Frequency

TI-55 III: Digital Root

Takes the digital root of an integer.  To find the digital root:

1.  Add up the number’s digits.

2.  The sum is over 10, add the digits again.

3.  Repeat step 2 until you get a single digit.

Or alternatively, use the formula dr(n) = 1 + ((n-1) mod 9) = n – 9 * integer((n-1)/9)

Program:

Partitions Allowed: 1-5, 1 register required

STEP	CODE	KEY	COMMENT
00	61	STO	Enter integer
01	00	0
02	75	-
03	09	9
04	65	*
05	53	(
06	53	(
07	71	RCL
08	00	0
09	75	-
10	01	1
11	54	)
12	55	÷
13	09	9
14	54	)
15	88	Intg
16	95	=
17	12	R/S	Display digital root

Input:  integer [RST] [R/S]

Result:  digital root

Test 1:  Input: 1555, Result: 7

Test 2:  Input: 38267, Result: 8

TI-55 III: Complex Number Multiplication

(a + bi)*(c + di) = (r1*r2) * e^(i*(θ1 + θ2))

Where r1 ∠ θ1 is the polar form of a + bi and r2 ∠ θ2 is the polar form of c + di.

Program:

Partitions Allowed:  2-4, 2 memory registers are required

STEP	CODE	KEY	COMMENT
00	52	X<>Y	Start with a
01	12	R/S	Prompt for b
02	41	INV
03	57	P-R	Convert to Polar
04	61	STO
05	01	1
06	52	X<>Y
07	61	STO
08	00	0
09	12	R/S	Prompt for c
10	52	X<>Y
11	12	R/S	Prompt for d
12	41	INV
13	57	P-R	Convert to Polar
14	61	STO
15	85	+
16	01	1	STO+ 1
17	52	X<>Y
18	61	STO
19	65	*
20	00	0	STO* 0
21	71	RCL
22	00	0
23	52	X<>Y
24	71	RCL
25	01	1
26	57	P-R	Convert to Rectangular
27	12	R/S	Display imaginary part
28	52	X<>Y
29	12	R/S	Display real part

Input:  a [RST] [R/S], b [R/S], c [R/S], d [R/S]

Result:  imaginary part of the product [R/S], real part of the product

Test 1:  (5 – 3i)*(4 + i)

Input: 5 [RST] [R/S], 3 [+/-] [R/S], 4 [R/S], 1 [R/S]

Result: -7 [R/S] 23   (23 – 7i)

Test 2: (-6 + 3i)*(2 + 2i)

Result:  -18 – 6i

TI-55 III:  Escape Velocity

v = √(2*G*m/r)

v = escape velocity (m/s)

G = University Gravitational Constant = 6.67384 * 10^-11 m^3/(kg*s^2)

m = mass of the planet (kg)

r = radius of the planet (m)

Note that 2*G = 1.334768 * 10^-10 m^3/(kg*s^2)

Program:

Allowed Partitions: 1-5

STEP	CODE	KEY	COMMENT
00	47	Eng	Set Engineering Mode
01	65	*	Start with mass
02	01	1
03	93	.	Decimal Point
04	03	3
05	03	3
06	04	4
07	07	7
08	06	6
09	08	8
10	42	EE
11	01	1
12	00	0
13	94	+/-
14	55	÷
15	12	R/S	Prompt for radius
16	95	=
17	13	√
18	12	R/S	Display escape velocity

Input:  mass (in kg) [RST] [R/S] radius (in m) [R/S]

Result: escape velocity (m/s)

Test 1:

Earth:  m = 5.97219 * 10^24 kg, r = 6.378 * 10^6 m

Input:  5.97219 [EE] 24 [RST] [R/S] 6.378 [EE] 6 [R/S]

Result:  ≈ 11.179E3 (11,179 m/s)

Test 2:

Jupiter:  m = 1.89796 * 10^27 kg, r = 71.492 * 10^6 m

Result:  ≈ 59.528E3  (52,528 m/s)

Eddie

This blog is property of Edward Shore, 2016