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This blog is property of Edward Shore. 2014.
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
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
: Radian
: 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:
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Planet/Celestial Object
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Mass (pounds)
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Avg. Radius (miles)
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Orbit (years)
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Earth
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1.317E25
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92.954E6
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1.000009092
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Jupiter
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4.186E27
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483.682E6
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11.86410888
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Pluto
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3.244E22
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3.67005E9
|
248.0906781
|