## Tuesday, May 29, 2018

### Fun With the Casio fx-3650P: Program Collection 5/29/2018

Fun With the Casio fx-3650P

Previous Entries

5/11/2014:

1. Circular Sectors
2. Stopping Sight Distance
3. Resistors in Parallel
4. Net Present Value
5. Rod Pendulum
6. Vectors: Dot and Cross Products

10/27/2015:

1.  Combination with Replacement
2.  Great Circle (Distance in km)
3.  Orbital Speed and Period
4.  Eccentricity and Area of an Ellipse
5.  Super Factorial
6.  Escape Velocity
7.  Finance: Payment of a Monthly Mortgage
8.  Wind Chill Factor
9.  Speed of Sound in Dry Air

7/2/2017:

1.  Modulus Function
2.  Normal CDF
3.  Sum:  Σ (AX + B)^C,  from X = 0 to X = Y
4.  Sun Altitude and Azimuth Based on the Vernal Equinox
5.  Trapezoid: Midsegment, Height, and Area
7.  General a list of X Random Integers from 0 to Y

Contents for this Blog Entry:

1. kth Derivative of f(x) = y^n
2. Sight Reduction Table
3. Distance Off at Second Bearing by Two Bearings and Run
4. Hydraulic Cylinder:  Force and Flow
5. Mass Held by Two Strings
6. Atwood Machine: Tension and Acceleration

kth Derivative of f(x) = x^n

This program calculates the kth derivative of x^n. Using the variables of the Casio fx-3650p, this program calculates:

‘d^A/dx^A X^B.  A is an integer, where B and X are real numbers.

Program (47 steps):
? → A : ? → B : → X : 1 → C : Lbl 1 : CB → C : B – 1 → B : A – 1 → A : A ≠ 0 Goto 1 : CX ^ B

Input order:  A = order, B = power, X = x

Examples:
A = 3, B = 2.5, X = 3.  Result:  1.08253175473
A = 4, B = 6, X = 1.  Result:  360

Sight Reduction Table

The program calculates altitude and azimuth of a given celestial body.

Inputs:
A: Local Hour Angle (LHA)
B: The observer’s latitude on Earth, north is positive, south is negative (L)
D: Declination of the celestial’s body, north is positive, south is negative (δ)

The latitude is often entered in degrees/minutes/seconds format.
Do this using the [ ° ‘ “ ] key.   The calculator is set in Degrees program.

Formulas:

Altitude:
H = asin (sin δ sin L + cos δ cos L cos LHA)

Zenith:
Z = acos ((sin δ – sin L sin H) ÷ (cos H cos L))
If sin LHA < 0 then Z = 360° - Z

Output variables:

C = altitude
X = zenith

Program (67 steps):
Deg : ? → A : ? → B : ? → D : sin^-1 ( sin D sin B + cos D cos B cos A ) → C cos^-1 ( ( sin D – sin B sin C ) ÷ ( cos C cos B ) ) → X: sin A > 0 360 – X → X : X

Source:  “NAV 1-19A Sight Reduction Table”    HP 65 Navigation Pac.  Hewlett Packard, 1974.

Distance Off at Second Bearing by Two Bearings and Run

This program calculates the distance off between to bearings.

Input variables:

A:  first bearing
B:  second bearing
C:  run (typically in miles, distance will be in the same length unit)

Distance formula:

D = C * sin A° / sin |A° - B°|

Since the Casio fx-3650P does not have an absolute value function, the workaround    (x^2) is used instead.

Program (31 steps):
Deg : ? → A : ? → B : ? → C : C sin A ÷ sin ( √ (A – B) ^ 2 ) → D

Example:
A = 15°, B = 8 °, C = 1.5 miles

Result:  3.185613024 miles
(updated 7/1/2018; the result is now correct)

Source:  Henry H. Shufeldt and Kenneth E. Newcomer.  The Calculator Afloat: A Mariner’s Guide to the Electronic Calculator Naval Institute Press:  Annapolis, MD.  1980

Hydraulic Cylinders: Force and Flow

The program calculates the force and flow.

Input variables:

X:  pressure
Y:  speed of the cylinder

Formulas:

Area: A = π (D^2 – C^2) / 4
Force: F = pressure * area = X * A
Flow:  q = velocity * area = Y * A

Program (36 steps):
? → D : ? → C : π ( D^2 – C^2 ) ÷ 4 → A : ? → X : AX ? → Y : AY

Example:

Input:
D = 8 in, C = 4 in  (8 in cylinder with a 4 in cutout), X = 68 psi, Y = 3.6 in/sec

Results:
Force:  2563.539605 lb,  Flow:  3.6 in^3/sec

Mass Held by Two Strings

The program calculates the tension of the two strings shown in the diagram above.  The tension of both strings are found by solving the following system of two equations:

cos A° * X – sin B° Y = 0
sin A° * X – cos B° * Y = M * g

Where g is Earth’s gravitation constant, or g = 9.80665 m/s^2 ≈ 32.174 ft/s^2

SI units are used in this program.

Program (45 steps):
Deg : ? → A : ? → B : ? → M : 9.80665 M ÷ ( tan A sin B – cos B ) → Y : Y sin B ÷ cos A → X Y

Example:

Input:
A = 43°, B = 89°, M = 16 kg

Results:
X = 234.4572201 N, Y = 171.4972763 N

Atwood Machine:  Tension and Acceleration

The program calculates the tension in the string of the Atwood Machine and the acceleration of the system.  If the acceleration is negative, the pulley is turning clockwise, otherwise it is turning counterclockwise.  The following system of equations is solved, where X is the tension of the strings and Y is the acceleration of the system.

X + A * Y = g * A
-X + B * Y = -g * B

Where g is Earth’s gravitation constant, or g = 9.80665 m/s^2 ≈ 32.174 ft/s^2

SI units are used in this program.

Program (45 steps):
? A : ? B : 9.80665 (A – B) ÷ ( A + B ) Y : A (9.80665 – Y) X Y

Examples:

Input:
A = 11.2 kg, B = 10.3 kg

Results:
X = 105.2367576 N, Y = 0.41051093 m/s^2

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

## Sunday, May 27, 2018

### HP Prime Colors (English and Spanish)

HP Prime Colors (English and Spanish)

Introduction

The program COLORES lets the user scroll through various colors (black/blanco, gray/gris, white/blanco, etc.) using the plus ( [+] ) and minus ( [ -] ) buttons.  End the program while pressing the escape button ([Esc]).

That’s all there is to it.  Enjoy this program.

HP Prime Program COLORES

drawnow(); // subroutine

EXPORT COLORES()
BEGIN
// 2018-05-27 EWS
// Colors in English and Spanish

// initialization
LOCAL i:=0,j:=1;
LOCAL lst:={{0,"black","negro"},
{#808080h,"gray","gris"},
{#FFFFFFh,"white","blanco"},
{#FF0000h,"red","rojo"},
{#800000h,"maroon","granate"},
{#FF00FFh,"magenta","magenta"},
{#400080h,"indigo","índigo"},
{#800080h,"purple","púrpura"},
{#000080h,"navy blue","azul marino"},
{#0000FFh,"blue","azul"},
{#00FFFFh,"cyan","cian"},
{#80D0FFh,"sky blue","cielo azul"},
{#003000h,"fir green","abeto verde"},
{#008000h,"green","verde"},
{#00FF00h,"lime","lima"},
{#FF8000h,"orange","naranja"},
{#FFFF00h,"yellow","amarillo"}
};
LOCAL s:=SIZE(lst);

RECT();
TEXTOUT_P("Press [+] or [-] to start.",
10,45,4);

LOCAL k,j1,j2,j3;
REPEAT
k:=GETKEY;

// plus key
IF k==50 THEN
i:=i+1; j:=(i MOD s)+1;
j1:=lst(j,1); j2:=lst(j,2);
j3:=lst(j,3);
drawnow(j1,j2,j3);
END;

// minus key
IF k==45 THEN
i:=i-1; j:=(i MOD s)+1;
j1:=lst(j,1); j2:=lst(j,2);
j3:=lst(j,3);
drawnow(j1,j2,j3);
END;

UNTIL k=4; // ESC key
END;

// subroutine
drawnow(x1,x2,x3)
BEGIN
RECT();
// polygon
FILLPOLY_P({{10,10},{10,150},{150,150},
{150,10}},x1);
// text
TEXTOUT_P("English:",160,20,4);
TEXTOUT_P(x2,160,40,4);
TEXTOUT_P("Español:",160,70,4);
TEXTOUT_P(x3,160,90,4);
END;

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

## Friday, May 25, 2018

### TI-84 Plus CE Programs from Casio FX-702P

TI-84 Plus CE Programs from Casio FX-702P

Notes

Here are several programs from the “Bilbliotheque De Programmes” for the Casio FX-702P translated for the Texas Instruments TI-84 Plus CE.  I understand that there is an English version of the book, but I have yet to find it online.  I am thankful to Google Translate since I am not fluent in French.

I named each of the programs FX702P##, starting with 02.  Of course, use the program name that fits.

Basic doesn’t translate directly to the TI-84 Plus programming language.  First of all, there are no line numbers for the TI-84 Plus, separate labels had to be created.  I added more descriptive lines to prompts and results strings because (1) the TI-84 Plus CE has a lot more programming space than the 1,680 bytes the Casio FX-702P offered, and (2) it helped me understand what the programs were.

The reference page ties to the page in the document, not the PDF page.

TI-84 Plus CE Program FX702P02  (ref. page 85)
Geometric Mean:  Enter 0 to stop data entry

"GEOMETRIC MEAN (85)"
"2018-05-20"
1→A:1→B
Lbl 20
Disp "X-",A
Input X
If X=0:Goto 60
A+1→A:B*X→B
Goto 20
Lbl 60
Disp "G=",B^(1/(A-1))

TI-84 Plus CE Program FX702P03 (ref. page 117)
Chemistry:  Analysis of an Acid

"ACID PRGM, 2018-05-20"
"(117)"
For(J,1,10)
Input "T= (0:END)",T
If T=0:Stop
37-T→T
Input "P*=",A
Input "PH=",P
Input "H=",H
.019T→B:A*10^(B)→A
Disp "P*=",A
P-.0146T→P:Disp "PH=",P
.0307A→C:C*10^(P-6.11)→D
Disp "(HC03­)=",D:Pause
D-(9.5+1.63H)(7.4-P)-24→C
(1-.0143H)C→B
Disp "BE=",B
(1+10^(P-6.11))→I:.0307AI→I
Disp "I=",I:Pause
End

TI-84 Plus CE Program FX702P04  (ref. page 131)
Accounting:  Calculate Accounts Receivable on outstanding invoices

"BILL REDUCTION (131)"
"2018-05-20 EWS"
1→Z:0→E:0→F
Lbl IN
"AMT: A"+toString(Z)+":"→Str1
Input Str1,A
"PERCENT: B"+toString(Z)+":"→Str1
Input Str1,B
"DAYS: C"+toString(Z)+":"→Str1
Input Str1,C
int(A*B/100*C/365+.5)→D
E+A→E:F+D→F
"AMT: D"+toString(Z)+":"→Str1:Disp Str1,D
Pause
Z+1→Z
Lbl NO
ClrHome
Disp "TOTAL INVOICES:",E
Disp "TOTAL INTEREST:",F
Input "COMMISSION:",X
E-F-X→N
Disp "NET RECIEVABLE:",N

TI-84 Plus CE Program FX702P05 (ref. page 73)
Thermal Stress

Disp "THERMAL STRESS"
"2018-05-22 EWS"
Disp "LINEAR EXPANSION","COEFFICIENT"
Input "L1:",L
Input "L2:",M
Disp "ELASTIC","COEFFICIENT"
Input "E1:",E
Input "E2:",F
Disp "CROSS SECTION"
Input "A1:",A
Input "A2:",B
Input "T:",T
EF(L-M)T→C
C/(EA+FB)→C
­(BC)→R:AC→S
ClrHome
Disp "THERMAL STRESS","A:","B:"
Output(2,5,R)
Output(3,5,S)

TI-84 Plus CE Program FX702P06  (ref. page 206)
Construction:  Concentrated Load on a Beam

Disp "BEAM FIXED AT","BOTH ENDS UNDER","CONCENTRATED LOAD"
"(63) 2018-05-22 EWS"
Degree
Input "YOUNG MOD (KG/MM^4):",E
Input "INERTIA (MM^4):",I
Input "LENGTH (MM):",L
E*I→B
Input "INTEREST POINT:",X
If X≤A
Then:A→Y:X→Z
Else:L-A→Y:L-X→Z
End
Z(L+2Y)→C:P(L-Y)^2/L^3→D
D*Z*Z/6/B*(C-3*Y*L)→F
tan^-1(D*Z/2/B*(C-2*Y*L))→S
D*(C-Y*L)→M
D*C/Z→W
ClrHome:Fix 4
Disp "FLEXION (MM):"," ANGLE:"," TORQUE:","SHELL LOAD:"
Output(1,14,F)
Output(2,14,S)
Output(3,14,M)
Output(4,14,W)
Float

TI-84 Plus CE Program FX702P07 (ref. page 41)
Electronics:  Active Low-Pass Filter

Disp "ACTIVE LOW-PASS FILTER"
"(41) 2018-05-22 EWS"
Input "FREQ (HZ):",F
Input "AF:",A
Input "C1 (MICRO-F):",C
C/1E6→C
√(2)/(4*A*π*F*C)→R
A*R→S
S/(A+1)→T
2*(A+1)*C*1E6→U
ClrHome
Disp "R1:","R2:","R3:",C2:"
Output(1,5,R)
Output(2,5,S)
Output(3,5,T)
Output(4,5,U)

Source: Casio.  “Bibliotheque De Programmes FX-702P” (French document)  1985

You can find a PDF of the book in French on this page: http://casio.ledudu.com/pockets.asp?type=53&lg=eng .  Scroll down to the bottom of the page.  Also, shout out to the casio.ledudu.com page because it is an excellent site!

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

## Saturday, May 19, 2018

### Update to the Big Font Math Reference

Original post:  https://edspi31415.blogspot.com/2018/04/april-11-2018-seven-years.html

I added several sections to the Big Math Reference document, which you can download on the right side of the blog.

* Days Between Dates Algorithm
*  Astronomy Formulas: Kepler's 3rd Law, Parallax, Luminosity, Schwartzchild Radius
*  Physics:  Projectile Motion, Linear Motion, Angular Motion
*  Electronics:  Ohm's Law, Resistance

Eddie

### Calculus/TI-84 Plus CE: Derivatives of kth Order

Calculus/TI-84 Plus CE:  Derivatives of Kth Order

Introduction

Can we find a general formula for a derivative of nth order?

d^n/dx^n f(x) = ?

The Power Function x^n

From calculus, we find that:

f(x) = x^n

d/dx x^n = n * x^(n - 1)

d^2/dx^2 x^n = n * (n – 1) * x^(n – 2)

d^3/dx^3 x^n = n * (n – 1) * (n – 2) * x^(n – 3)

d^4/dx^4 x^n = n * (n – 1) * (n – 2) * (n – 3) * x^(n – 4)

Note that for order k,

d^k/dx^k x^n = (n * (n – 1) * (n – 2) * (n – 3) * … * (n – (k – 1)) ) * x^(n – k)

d^k/dx^k x^n = (n * (n-1) * (n-2) * … * 1)/((n-k) * (n-k-1) * … * 1) * x^(n – k)

d^k/dx^k x^n = n! / (n – k)! * x^(n – k)

With the gamma function property Γ(z + 1) = z!,

d^k/dx^k x^n = Γ(n + 1)/Γ(n – k + 1) * x^(n – k)

The above formula allows us to calculate the kth derivative of x^n, even when k is not an integer.

TI-84 Plus CE Program NDERPOW

"2018-05-18 EWS"
Disp "D^K/DX^K X^N"
Input "POWER (N):",N
Input "VALUE    :",A
Input "ORDER (K):",K
If N≥K
Then
N!/(N-K)!*A^(N-K)→D
Else
0→D
End
Disp D

The program NDERPOW calculates the numerical derivative of d^k/dx^k x^n.  For this particular program, k must be an integer since non-integers are not accepted on the TI-84 Plus’ factorial function.

The Exponential Function e^(a*x), where a is a constant

f(x) = e^(a*x)

d/dx e^(a*x) = a * e^(a*x)

d^2/dx^2 e^(a*x) = a^2 * e^(a*x)

d^3/dx^3 e^(a*x) = a^3 * e^(a*x)

d^4/dx^4 e^(a*x) = a^4 * e^(a*x)

With the order k…

d^k/dx^k e^(a*x) = a^k * e^(a*x)

Just like the last case, k does not have be an integer.

TI-84 Plus CE Program NDEREXP

"2015-05-18 EWS"
Disp "D^K/DX^K e^(A*X)"
Input "COEFF (A):",A
Input "VALUE    :",X
Input "ORDER (K):",K
A^K*e^(A*X)→D
Disp D

The program NDEREXP calculates the numerical derivative of d^k/dx^k e^(a*x).

The Sine Function sin(a*x) and the Cosine Function cos(a*x)

We are working radian angle measure.

f(x) = sin(a*x)

d/dx sin(a*x) = a * cos(a*x)

d^2/dx^2 sin(a*x) = -a^2 * sin(a*x)

d^3/dx^3 sin(a*x) = -a^3 * cos(a*x)

d^4/dx^4 sin(a*x) = a^4 * sin(a*x)

Notice a pattern, alternating between sin and cos.  To the kth order (k is an integer),

d^k/dx^k sin(a*x) =

(-1)^int(k/2) * a^k * cos(a*x), when k is odd

(-1)^int(k/2) * a^k * sin(a*x), when k is even

If we put the piecewise function into one statement:

d^k/dx^k sin(a*x) = (-1)^int(k/2) * a^k * ( 2*frac(k/2)*cos(a*x) + 2*frac((k+1)/2)*sin(a*x) )

Note that:

2 * frac(k/2) = 1 for all odd integers k, 0 for all even integers k

Likewise, 2 * frac((k + 1)/2) = 0 for all odd integers k, 1 for all even integers k

And, (-1)^int(k/2) produces a pattern of 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, …

TI-84 Plus CE Program NDERSIN

"2018-05-19 EWS"
Disp "D^K/DX^K sin(A*X)"
Input "COEFF (A):",A
Input "VALUE    :",X
Input "ORDER (K):",K
(­1)^iPart(K/2)*A^K*(2*fPart(K/2)*cos(A*X)+2*fPart((K+1)/2)*sin(A*X))→D
Disp D

Similarly,

f(x) = cos(a*x)

d/dx cos(a*x) = -a * sin(a*x)

d^2/dx^2 cos(a*x) = -a^2 * cos(a*x)

d^3/dx^3 cos(a*x) = a^3 * sin(a*x)

d^4/dx^4 cos(a*x) = a^4 * cos(a*x)

Likewise:

Notice a pattern, alternating between sin and cos.  To the kth order (k is an integer),

d^k/dx^k sin(a*x) =

(-1)^int(k/2 + 1/2) * a^k * sin(a*x), when k is odd

(-1)^int(k/2 + 1/2) * a^k * cos(a*x), when k is even

If we put the piecewise function into one statement:

d^k/dx^k cos(a*x) = (-1)^int(k/2 + 1/2) * a^k * ( 2*frac(k/2)*sin(a*x) + 2*frac((k+1)/2)*cos(a*x) )

TI-84 Plus CE Program NDERCOS

"2018-05-19 EWS"
Disp "D^K/DX^K cos(A*X)"
Input "COEFF (A):",A
Input "VALUE    :",X
Input "ORDER (K):",K
(­1)^iPart((K+1)/2)*A^K*(2*fPart(K/2)*sin(A*X)+2*fPart((K+1)/2)*cos(A*X))→D
Disp D

NDERSIN and NDERCOS are numeric derivatives for sine and cosine, respectively.  Note for NDERSIN and NDERCOS, the calculator is set to Radian mode, and K (order) should be an integer.

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

## Saturday, May 12, 2018

### Retro Review Casio fx-702P

Retro Review Casio fx-702P

General Information

Company:  Casio
Type:  Scientific, BASIC Programmable
Memory:  26 memory registers, can be expanded to 226 memory registers;
1,680 programming steps maximum
Battery:  2 x CR 2032
Years:  1981 - 1984
Original Cost: approximately \$200 (from the research I could gather)
Display:  20 characters with a contrast wheel

I picked up a Casio fx-702P at a local monthly swap meet in Pasadena, which came along with a Casio FX-2 cassette interface and a FP-10 printer.  Not bad for \$15, which normally this would sell anywhere from \$80 to \$150 on eBay.

My review will focus on the calculator.

Features

The Casio fx-702P is a landscape BASIC programmable calculator designed for storing up to 10 programs.  The maximum amount of steps allocated is 1,680; which seem quite small compared to today’s computers.

The set of scientific functions is nice: hyperbolic functions, linear regression (in the form of y = a + b*x), sign function, integer and fraction parts.

Programs on the fx-702P can be printed using the FP-10 printer.  As of this review I have yet to test the printer, other than providing it with four AA batters, the printer does turn on.  Programs can also be stored to cassette tapes via the FX-2.  I am assuming the connectors on the FX-2 can be connected to any compatible cassette recorder.

The Programming Language

What I find unique about the programming language of the Casio fx-702P is the language.  A lot of the commands are three characters, for example:

 Command usually used in BASIC Casio fx-702P command INPUT INP GOSUB GSB PRINT PRT RETURN RET SQRT SQR ASIN ASN

The DMS allows results to be displayed in a degrees°minutesseconds” format.  DEG(degrees, minutes, seconds) converts DMS to decimal.

A nice command is the SET command, where you can set the number of decimal places to be displayed.  For example SET F4 sets the calculator to FIX 4 mode.

Changing the angle measurement is interesting, you have to type in MODE and the specific number: MODE 4 for degrees, MODE 5 for radians, and MODE 6 for gradients.

Subroutines can work within and between programs.

Arrays and lists can be used, however, they come at an expense of an already low amount of programming steps.  There is only one array and list name, which is A.  A(i) first maps to the other one-letter variable names B-Z first.

What surprises me the most is the lack of ROM or RAM cards for the fx-702P.  There is no way to expand the number of steps on the fx-702P.

Keyboard

On the surface, the Casio fx-702P looks like a good, clean keyboard.  The keys are pretty responsive.  The F1 (red) and F2 (blue) shift keys have a nice contrast.

However, with the batteries I used, which were purchased from a swap meet, the calculator only lasted a couple days (total use I estimate to be an hour in those days) before the display went blank and the keys became unresponsive. Fortunately, using brand name batteries, like Duracell provides better and steady results.

Verdict

The Casio fx-702P is a good programming calculator and I understand why it is highly sought.  If you want to own one, I would look for the best price.

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

### HP Prime: Solving Projectile Motion with the Solve App

HP Prime:  Solving Projectile Motion with the Solve App

This blog entry is a demonstration using the Solve App to solve a variety of problems regarding projectile motion.

Getting Started

Go to the Solver App by pressing [Apps] and selecting the Solve application.  If need be, clear all the equations by pressing [Shift] [ Esc ] and selecting (OK) at the Clear All Expressions prompt.

Enter the following equations:

E1:  X = V*COS(θ)*T
E2:  Y = V*SIN(θ)*T – 0.5*G*T^2

Variables:

X = distance in the x direction
Y = distance in the y direction
Θ = angle the projectile is shot
V = initial velocity
T = number of seconds
G = Earth’s gravitation constant:  9.80665 m/s^2, 32.174 ft/s^2

Right now, we are assuming that there is no acceleration in the X direction, the projectile starts at point (0,0), and there is no air resistance.

Finally, put the calculator in Degrees mode, which the angle indicator will display °.

Example 1:  Determining Distance and Time it takes for the Projectile to complete

Given the following data:

θ = 40°
G = 9.80665 m/s^2
V = 35 m/s
Y = 0

Select X and T to solve.  Pressing (Solve) will give the following results:

X = 0 and T = 0.  Well, that went well.  We have an obvious answer.  What we are looking for is the distance traveled after the projectile has been shot.  Fortunately, the HP Solve app allows us to provide initial guesses.

Let’s put an initial guess of T = 100 and press (Solve) again.  We now get the following results:

X = 123.017492971
T = 4.58822662969

The projectile will travel approximately 123 m in the x direction and it will take slightly after four and half seconds before impact.

Example 2:  Determining initial velocity (V) and angle (θ)

Given the following data:

X = 125.15 m
Y = 3.10896 m
T = 5 s
G = 9.80665 m/s^2 (SI units are used)

The projectile landed on a hill 125.15 m away with an elevation 3.10896 m (10.2 ft) higher than the starting point.

You can clear the data variables with [Shift] [Esc].  Since you are solving for V and θ this time, make sure only those variables are checked.

Select θ, give it an initial value of 0, and press (Solve).

The results are:

θ = 45.1238194711
V = 35.474510698

Example 3:  Hitting a Moving Target

Find the required initial velocity (V) to land a projectile on a wagon moving at rate of 7.33 ft/s.  The wagon is 5.5 ft high.  The wagon is 20 ft away when the projectile is shot.  Assume the projectile is shot at 30°.

The first step is to modify equation E1.  Since the X distance contains a moving object, adjust the left side to 20 + 10*T.   20 for the initial distance plus 10*T for the moving wagon.

The equations will be set as such:

E1:   20+10*T = V*COS(θ)*T
E2:   Y = V*SIN(θ)*T – 0.5*G*T^2

Next enter the following variables:

θ = 30
Y = 5.5 (height of the wagon)
G = 32.174 (we are using US units)

Note that there only five variables this time.  Select V and T to solve for.  We want T selected because we don’t know what T is.

Select T and enter 1 as an initial guess (actually, we really want a small positive number should do it, as long as T is not zero).  .

We get a solution of:

T = 0.818269246898
V = 39.7700027103

This means we should fire the projectile at about 39.77 ft/s and it will land on the wagon.

Example 4:  The Maximum Height of a Projectile

What is the maximum height of a projectile found for a projectile shot at 36° at 54 ft/s?  How far (horizontal distance) does the projectile would have traveled to get that point?

For this problem, we are going to use three equations but they will be solved separately.  Set the equations as such:

E1:  X = V*COS(θ)*T
E2:  Y = V*SIN(θ)*T – 0.5*G*T^2
E3:  0 = V*SIN(θ) – G*T

The projectile reaches its height when the rate of the change in the y direction is 0.  In calculus terms, when Y’(T) = 0 where Y’(T) = dY/dT = V*SIN(θ) – G*T.

On the Symb screen, select only E3.  Press [Num] and enter the variables:

V = 54
θ = 36
G = 32.174 (ft/s^2)

Solve for T.  The result is: T = 0.76305137487.

It takes the projectile about 0.7630 seconds for the projectile to reach its height.

The next step is press [Symb] and select the equations E1 and E2.  Now press [Num], select X and Y to solve for and press (Solve).

The solution is:

X = 43.0981662447
Y = 15.6563253279

The distance is about 43.0982 ft and the height is 15.6563 ft.

Eddie

All original content copyright, © 2011-2018.  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.  Please contact the author if you have questions.

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