(12C) Projectile Motion

01072017, 05:32 PM
Post: #1




(12C) Projectile Motion
HP 12C: Projectile Motion, No Air Resistance: Maximum Distance (U.S. Units)
For an object that travels in a projectile motion, we can track its range (distance traveled from the beginning) and height by: R = v^2 * sin (2 * θ)/g H = (v^2 * (sin θ)^2) / (2 * g) Where: v = initial velocity θ = initial angle g = Earth’s gravity. For in US units, g = 32.1740468 ft/s^2. This program uses the approximation g ≈ 32.174 ft/s^2 The projectile will have maximum range (distance) if we aim the object at 45°. Aside: Why? Let’s let range (R) be a function of angle (θ): R = v^2/g * sin(2 * θ) Find the critical points by finding the zero of the first derivative: dR/dθ = 2 * v^2/g * cos (2 * θ) 0 = 2 * v^2/g * cos (2 * θ) 0 = cos (2 * θ) arccos 0 = 2 * θ π/2 = 2 * θ θ = π/4 Now we can use the second derivative to test whether the function is at a maximum (less than 0) and minimum (more than 0) at the crucial point. d^2 R/dθ^2 = 4 * v^2/g * sin(2 * θ) Let θ = π/4 4 * v^2/g * sin(2 * π/4) = 4 * v^2/g * sin(π/2) = 4 * v^2/g < 0 (We are assuming the initial velocity is positive, and g ≈ 32.174 >0) Since the second derivative at θ = π/4 is negative, the range is at its maximum. Note that calculus is done when the angles are measured in radians. π/2 in degrees is 90° and π/4 in degrees is 45°. (We are only concentrating on angles between 0° and 90°) To find the maximum range and height, substitute at θ = 45° and range and height are: R = v^2 /g H = v^2 / (4 * g) The time this certain projectile lasts is: T = (v * √2) / (2 * g) Program: Keep in mind: this is done on the HP 12C (regular). For the HP 12C Platinum, the code for Last X is 43, 40 ([ g ] [ + ]) STEP; KEY; CODE NUMBER 01; STO 1; 44, 1 02; 2; 2 03; ÷; 10 04; LST x; 43, 36 05; √x; 43, 21 06; *; 20 07; 3; 3 08; 2; 2 09; .; 48 10; 1; 1 11; 7; 7 12; 4; 4 13; STO 0; 44, 0 14; ÷; 10 15; R/S; 31 16; RCL 1; 45, 1 17; ENTER; 36 18; *; 20 19; RCL 0; 45, 0 20; ÷; 10 21; R/S; 31 22; 4; 4 23; ÷; 10 24; GTO 00; 43, 33, 00 Input: velocity in ft/s (convert from mph to ft/s by multiplying it by 22/15) Output: time of projectile in seconds, [R/S] range of projectile in feet, [R/S] height of projectile in feet Example: V = 25 mph = 36.6666667 ft/s (110/3) Output: Time: 0.81 sec, Range: 41.79 ft, Height: 10.45 ft Source: Rosenstein, Morton. Computing With the Scientific Calculator Casio: Tokyo, Japan. 1986. ISBN10: 1124161430 

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