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.

Quick Review: Casio fx-9910CW 2nd Edition

Quick Review: Casio fx-9910CW 2^nd Edition

The Casio fx-9910CW 2^nd Edition is an update of the fx-991CW first released in 2023.

Product Pages

United States:

https://www.casio.com/us/scientific-calculators/product.FX-9910CW/

The fx-9910CW has two keyboards: black with gold font and pink with purple font (limited). I have one each and Casio has improved on the readability on the pink edition with the dark purple font.

I’m sure this model will be available world wide soon.

What’s the Same and What’s Different

Overall, the fx-9910CW 2^nd Edition has the same mathematical features as the fx-991CW. The modes included are:

Calculate: the main app for mathematical calculations

Statistics: 1 and 2 variable with seven regression models:

Linear:  y=a+bx

Quadratic: y=a+bx+cx^2

Logarithmic:  y=a+b*ln(x)

Exponential:  y=a*e^(bx)

Power I:  y=a*b^x

Power II: y=a*x^b

Inverse:  y=a+b/x

Statistical graphs may be generated with the QR feature.

Distribution: Binomial, Normal, and Poisson distributions, along with their inverses. The functions work with lower-tail probabilities (-∞ or 0 to x).

Spreadsheet: 5 columns, 45 rows. 2,380 byte memory.

Table: Generate a table for up to two functions f(x) and g(x). Generate graphs with the QR feature.

Equations: Linear systems (up to 4 x 4), polynomials up to 4^th order, and a general equation solver.

Inequalities: Solve inequalities for polynomials up to the 4^th order.

Complex Numbers: Complex number arithmetic with polar/rectangular conversion, integer powers, real/imaginary parts, conjugate

Base N: Base mode for the traditional decimal, hexadecimal, octal, and binary basis. Binary integers are up to 31 bytes with one sign bit.

Matrix: Four matrices, up to size 4 x 4, with basic matrix functions such as determinant, transpose, and inverse.

Vector: Four vectors, 2D or 3D, with dot product, cross product, and norm.

Math Box: Dice Roll and Coin Toss simulations

47 scientific constants (SI units) and unit conversions

We still have the newer Casio style of navigation keys, the Home Key, the Back Key, the Settings key, and the Scroll up and down keys.

What is different?

First, the fx-9910CW is not solar powered, but instead runs on a single AAA battery.

Welcome changes:

The menus all have short cut keys! This eliminates the requirement to scroll down menus, which some get long, to find the sub menus and functions. This makes the operating the calculator a lot easier and more efficient, eliminating additional key strokes by repeatedly pressing the arrow keys.

For example, to get the absolute value in Calculate mode, press [ CATALOG ], [ 3 ] for Numeric Calc, [ 1 ] for Absolute Value.

In the same mode, we call up the Speed of Light constant (c), press [ CATALOG ] , [ 7 ] for Sci Constants, [ 1 ] for Universal, and [ 3 ] for c.

There is a setting to turn the hide the shortcut numbers but thankfully it won’t turn off the ability to navigate by using short cut keys.

The 10^[] and FORMAT keys return to familiar form! Like the fx-CG 100, we have an option on how the 10^[] and FORMAT keys operate.

10^[] Key:

(1) Power: Acts like a power key like the fx-991CW. I read that there are potential problems with calculating with this key leaving calculations to return unexpected answers.

(2) Sci. Notat: This returns this key to the traditional scientific notation key. When this option is selected, the 10^ is shown in a small font. This key now acts like the [ EE ] or [ EXP ] keys on other scientific calculators. This is the default setting.

FORMAT Key:

(1) -π√ ←→ Decimal. Pressing the FORMAT key just toggles between decimal and the exact (when available) format of answers. This brings back the beloved [ S←→D ] key. This is the default settings.

(2) Format Menu. This is the format menu that is presented, similar to the fx-991CW. We trade off the quick toggle for additional formatting options.

Shortcut Catalog. Pressing [ SHIFT ] [ CATALOG ] is a new feature and provides a listing of all mathematical functions available in the active mode. Up to 15 functions are shown, using the arithmetic keys (+, -, ×, ÷) and the decimal point (.) as additional shortcut keys. The shortcut catalog does not have the conversions or constants.

These changes are welcome and enhance the operating experience of the calculator.

My next wish is that the fx-9910CW included the Algo mode (algorithm mode) that allowed small programs (via a mix of Scratch and Casio Basic). You can find out more details here:

https://edspi31415.blogspot.com/2025/08/casio-fx-92-college-plotting-lines.html

I just think the algorithm mode is neat, gives students an introduction to programming, and would have fit the fx-9910CW well.

Great job, Casio. I think Casio listened to calculator users and brought back these improvements, and brought it more in line with operating the fx-CG 100/Graph Math Plus. If you were hesitant about the fx-991CW, the fx-9910CW 2^nd Edition is a good time to jump in.

Eddie

All original content copyright, © 2011-2025. 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.

HP 71B Programs: September 2025

HP 71B Programs: September 2025

One of my favorite calculators/pocket calculators of all time is the HP 71B.

ADDMOD: (a + b) mod n = a mod n + b mod n

10 DESTROY A, B, N

20 DISP “(A+B) MOD N” @ WAIT .5

30 INPUT “A? “; A

40 INPUT “B? “; B

50 INPUT “C? “; C

60 S = MOD(A, N) + MOD(B, N)

70 S = MOD(S, N)

80 DISP “SUM = “; S

MULTMOD: (a * b) mod n = a mod n * b mod n

10 DESTROY A, B, N

20 DISP “(A*B) MOD N” @ WAIT .5

30 INPUT “A? “; A

40 INPUT “B? “; B

50 INPUT “N? “; N

60 P = MOD(A, N) * MOD(B, N)

70 P = MOD(P, N)

80 DISP “PRODUCT = “; P

Examples:

A	630	48	15
B	320	99	47
N	700	15	7
SUM:	250	12	6
PRODUCT:	0	12	5

GABLE: Area of a Gable Roof

10 DESTROY P, S, L

20 DEGREES

30 INPUT “PITCH (IN.)? “; P

40 INPUT “SPAN (FT.)? “; S

50 INPUT “LENGTH (FT.)? “; L

60 A = 2 * S * L / COS(ATAN(P/12))

70 DISP “AREA = “; A; “ FT^2”

Input: Pitch (P)	6”	3”	4”
Input: Span (S)	110” (110/12 ft)	5’ 8” (5 + 8/12 ft)	15 ft
Input: Length (L)	100” (100/12 ft)	10’	10 ft
Output: Area (A)	170.81074828 ft^2	116.821326059 ft^2	316.227766017 ft^2

INTENSE: Light Intensity of a Spherical or a Cylindrical Light Source

intensity = power / surface area

intensity (W/m^2), power (W), surface area (m^2)

Surface area: sphere = 4 * π * r^2, cylinder = 2 * π * (r * h + r^2)

10 DESTROY P, K$, R, H

20 DISP “INTENSITY OF LIGHT!” @ WAIT .5

30 INPUT “POWER (W)? “; P

40 DISP “1. SPHERE 2. CYLINDER”

50 DELAY 0,0

60 K$ = KEY$

70 IF K$ = “1” OR K$ = “S” THEN GOTO 100

80 IF K$ = “2” OR K$ = “C” THEN GOTO 200

90 GOTO 40

100 INPUT “RADIUS (M)? “; R

110 A = 4 * PI * R^2

120 GOTO 300

200 INPUT “RADIUS (M)? “; R

210 INPUT “HEIGHT (M)? “; H

220 A = 2 * PI * (R * H + R^2)

230 GOTO 300

300 I = P / A

310 DISP I; “W/M^2”

Examples:

Sphere: r = 4 m, Power = 60 W: Intensity = .298415518297 W/M^2

Cylinder: r = 1.25 m, h = 0.75 m, Power = 70 W; Intensity = 4.45633840656 W/M^2

SIMPLE: Simple Interest – Banker’s Rule

interest = amount * rate% / 100 * #days / 360

final = principal + interest (final, maturity value)

10 DESTROY P, R, I, M

20 DISP “SIMPLE INTEREST” @ WAIT .5

30 INPUT “AMT? “; P

40 INPUT “RATE%? “; R

50 INPUT “# DAYS? “; D

60 I = P * R * D / 360 / 100

70 M = P + I

80 IMAGE K, M10D.2D

90 DISP USING 80; “INT.=”, I @ PAUSE

100 DISP USING 80; “FINAL=”, M

Notes:

* When the HP 71B is in pause, press [ f ] [ + ] to continue (CONT).

* To stop execution, press [ ON ] (ATTN).

* IMAGE K, M10D.2D: display “prompt string”, number rounded to 2 decimal places. The number is right-justified, make the prompt string 7 characters or less to fit both the string and number on the screen.

Examples

Input: Amount	1500	2000	2800
Input: Rate %	4.00%	8.00%	6.65%
Input: # Days	180	90	60
Output: Interest	30	40	31.03
Output: Final	1530	2040	2831.03

AIRSOUND: Speed of Sound in Air

S = 331.5 + .606 * T

S = speed of sound in air, m/s

T = temperature in °C

10 DESTROY K$, S, T

20 DISP “SPEED OF SOUND IN AIR” @ WAIT .5

30 DISP “SOLVE FOR…” @ WAIT .5

40 DISP “1. SPEED 2. TEMP”

60 K$ = KEY$

70 IF K$=”1” OR K$=”S” THEN GOTO 100

80 IF K$=”2” OR K$=”T” THEN GOTO 200

90 GOTO 40

100 INPUT “TEMP (°C)? “; T

110 S = 331.5 + .606 * T

120 DISP “S=”; S; “ M/S”

130 END

200 INPUT “SPEED (M/S)? “; S

210 T = (S- 331.5) / .606

220 DISP “T=”; T ; “°C”

230 END

Examples:

Input: T = 74 °F = (32 + 1/3) °C Output: S = 345.64 m/s	Input: S = 350 m/s Output: T = 30.5280528053 °C
Input: T = -8.5 °C Output: S = 326.349 m/s	Input: S = 382 m/s Output: T = 83.3333333333 °C

Eddie

All original content copyright, © 2011-2025. 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.

The author does not use AI engines and never will.

The Product Formula Chart Aid: P = A × B

The Product Formula Chart Aid: P = A × B

A Learning and Memorization Aid

A lot of mathematical formulas, basic relationships in physics and other applications are often in the form of:

P = A × B

where P is the product of two factors, A and B.

Examples include:

Distance: distance = velocity × time

Ohm’s Law: power = current × voltage

Newton’s Second Law: force = mass × acceleration

A chart in a shape of a circle (some people use a triangle) can be used to illustrate the relationship between the three variables.  I see charts of this time in various math books and videos applied to many applications.  An example is Ohm's Law as illustrated by Electrician U (skip to https://youtu.be/-oHzc_DbaGw?t=17).

Going across means multiply, while vertically means divide.

P = A × B

A = P ÷ B

B = P ÷ A

Hope you find this helpful,

Eddie

Source:  

Electrician U.  "5 Formulas Electricians Should Have Memorized!"  March 15, 2023.   https://www.youtube.com/watch?v=-oHzc_DbaGw.  Accessed May 27, 2025

All original content copyright, © 2011-2025. 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.

All posts are 100% generated by human effort.  The author does not use AI engines and never will.