## Sunday, December 28, 2014

### TI-84+: Doppler Effect, Finding an Equation of a Line with 2 Points, Arc Length of f(x), Orbit around the Sun

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Doppler Effect for Sound in Air

Program DOPPLER

: Input “TEMP IN ⁰F:”, T

: Input “SOURCE FREQ (HZ):”,F

: 331.4+0.6*5/9*(T-32)→C

: Disp “VELOCITY”, “(TOWARDS IS NEGATIVE)”

: Input “(MPH):”, S

: S/2.2369→S

: C/(C+S)*F→R

: Disp “SPEED OF SOUND (MPH)”, C*2.2369

: Disp “OBSERVED FREQUENCY”, R

(223 bytes)

Variables:

C = calculated speed of sound in air, stored in m/s

S = speed of observer relative to the source, stored in m/s.  S<0 if the source and observer are getting closer.  S>0 when the source and observer are moving apart.

F = source frequency, in Hz

R = observed frequency, in Hz

Example:

An observer is moving towards emitting a sound of 261.6 Hz (Middle C) at about 1.05 mph.  Input velocity as -1.05.

Results:

Speed of Sound:  769.6427267 mph

Observed Frequency: 261.9573804 Hz

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Finding the Equation of a Line between Two Points

Enter two points as complex numbers (x+y*i).  The complex value i is accessed by pressing [2nd] [ 3 ]. Direction is important, P is your starting point and Q is your finishing point.

Program PT2LINE

: a+bi

: Disp “ENTER POINTS AS”,”COMPLEX NUMBERS”,”X+Yi”

: Prompt P, Q

: angle(Q-P)→S

: If S=90⁰ or S=-90⁰

: Then

: Disp “X=”, real(P)

: Else

: imag(P)-tan(S)*real(P)→C

: Disp “Y=”,tan(S),”* X +”,C

: End

(143 bytes)

Example 1:

Find an equation for the points, going from (2,5) to (4,6).

Input:

P = 2+5i

Q = 4 + 6i

Output:

Y = 0.5 * X + 4

Example 2:

Find an equation for the points, going from (7,3) to (9,-1).

Input:

P = 7+3i

Q = 9-i

Output:

Y = -2 * X + 17

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Arc Length of a Function

Formula: S = ∫(√(1+f’(x)^2) dx, a, b)

We will use a trick with the function variable to pull this integral off.

** Caution: This will provide an approximate answer.  Best use four to five digits at most.

Note: Enter Y1 by pressing [VARS], [ right ] for Y-VARS, [ 1 ] for Function, [ 1 ] for Y1.

Program ARCLENGT

: Disp “INPUT Y1 AS A STRING”

: Prompt Y1, A, B

: fnInt(√(1+nDeriv(Y1,X,X)^2),X,A,B)→S

: Disp “ARC LENGTH:”,S

(91 bytes)

Example:  Find the arc length of the function y = e^(-x^2/2) from x = 0 to x = 3.

Input:

Y1:  “e^(-X^2/2)”   (you can leave the second quote out)

A:  0

B:  3

Result:

S:  3.20863  (approximately)

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Orbit around the Sun Using Kepler’s Third Law

(using US Units)

The Mass of the Sun is 4.384 x 10^30 pounds.

The universal gravitational constant is:

G = 6.67384 x 10^-11 m^3/(s^2*kg) = 7.23243 x 10^-6 mi^3/(y^2*lb)

Kepler’s third law is:

P = √((4*π^2*a^3)/(G*(m1 + m2))

Where

a = average radius between the two objects

G = universal gravitational constant

m1 = mass of object 1

m2 = mass of object 2

P = orbit of object 2 around object 1

Letting m1 be the mass of the sun and simplifying in U.S. units, the periodic orbit simplifies to:

P ≈ √((5458527.439*a^3)/(4.3840 x 10^30 + m2))

** Approximate values

Program KEPLER3

: Disp “ORBIT AROUND THE SUN”

: Input “MASS (POUNDS):”, M

: Input “AVG. RADIUS (MI):”, A

: √((5458527.439*A^3)/(4.384E30+M))→P

: Disp “ORBIT IN YEARS:”,P

(135 bytes)

Examples:

 Planet/Celestial Object Mass (pounds) Avg. Radius (miles) Orbit (years) Earth 1.317E25 92.954E6 1.000009092 Jupiter 4.186E27 483.682E6 11.86410888 Pluto 3.244E22 3.67005E9 248.0906781

This blog is property of Edward Shore.  2014.