arla-phy.htm
Definitions
A = Cross Sectional Area of a Tube (square inches = in.in) a = Acceleration (feet per second squared = ft/sec.sec) f = Force (pounds force = lbf) p = Gas Pressure (pounds force per square inch = lbf/in.in = psi) r = Radius of a Tube (inches = in) s = Distance (feet = ft) t = Time (seconds) v = Velocity (feet per second = ft/sec) fps = ft/sec Pi = Circumference of a Circle/Diameter (3.14) sqrt = square root sin = trigonometric sin function Note: A period is often used here to denote multiplication because of the limitations of ASCII. I.E., (sec.sec) is the same as (seconds x seconds) or (seconds squared).
Basic relationships
s = v.v/2a a = v.v/2s v = sqrt(2as) v = at t = v/a a = v/t s = vt/2 v = 2s/t t = 2s/v f = pA A = f/p p = f/A A = Pi.r.r D = 2r
Distance to Accelerate From Rest to a specific velocity (related to tube length)
s = v.v/2a ex v = 2,000 ft/sec a = 10 g (322 ft/sec.sec) s = (2,000 ft/sec)x(2,000 ft/sec) ----------------------------- = 3,105 ft (2)x(322 ft/sec.sec) Table of Distances (assuming no drag) Accel Velocity (fps) (g) 400 800 1,200 1,600 2,000 1 2,484 9,938 22,360 39,752 62,112 3 828 3,312 7,453 13,250 20,704 10 248 994 2,236 3,975 6,211 20 124 497 1,118 1,988 3,106 50 50 199 447 795 1,242 100 25 99 224 398 621 200 12 50 112 199 310 500 5 20 45 80 124 Distance to accelerate from one velocity to another s = s1 - s0 s1 = distance to accelerate to final velocity s0 = distance to accelerate to initial velocity (assuming the same acceleration) ex s1 = v1.v1/2a v1 = 5,000 ft/sec a = 96.6 ft/sec.sec (3 g) s1 = 129,400 ft s0 = v0.v0/2a v0 = 2,000 ft/sec a = 96.6 ft/sec.sec (3 g) s0 = 20,700 ft s = s1 - s2 = 129,400 ft - 20,700 ft = 108,700 ft Notes: 1. This means that to accelerate a launch vehicle from rest to 2,000 fps at 200 g the tube needs to be 310 ft long. If you are going to launch people using a horizontal tube using 3 g the tube would be 20,704 ft long. 2. This calculation can be used to approximate the altitude required for the ramjet to accelerate to a given velocity. I.E., if the launch vehicle leaves the tube at 2,000 fps and accelerates at only 3 g then it will not reach 5,000 fps until 108,700 ft altitude. A lower ramjet acceleration would probably not achieve 5,000 fps before running out of air. 3. These calculations work for deceleration as well. For vertical launches use the velocity at flameout and 32.2 as the acceleration to get the coast altitude (assuming no drag). Vertical coast distance when launched at an angle other than vertical (assuming no drag). s = (v.v/2a) where v = sin(theta).(v) ex s = distance in the vertical direction theta = 60 degrees (from the horizon) sin(60 deg) = 0.866 v = (0.866).(5,000 ft/sec) = 4,330 ft/sec a = 32.2 ft/sec.sec s = (4.330 ft/sec).(4.330 ft/sec) ------------------------------------- (2).(32.2 ft/sec.sec) = 291,000 ft or 55 miles Vertical coast distance from an initial altitude s = s0 + s1 s0 = initial altitude s1 = additional coast altitude ex s0 = 100,000 ft s1 = 291,000 ft s = 391,000 ft
Velocity of a launch vehicle given an acceleration and time
v = at ex a = 32.2 ft/sec t = 1 sec v = (32.2 ft/sec.sec)x(1 sec) = 32.2 ft/sec Table of Velocities (ft/sec) Time (sec) a (g) 1 2 3 5 10 20 50 100 1 32 64 97 161 322 644 1,610 3,220 3 97 193 290 483 966 1,932 4,830 9,660 5 161 322 483 805 1,610 3,220 8,050 16,100 10 322 644 966 1,610 3,220 6,440 16,100 32,200 20 644 1,288 1,932 3,220 6,440 12,880 32,200 64,400 50 1,610 3,220 4,830 8,050 16,100 32,200 80,500 161,000 Notes: 1. This can also be used to determine how long the ramjet needs to operate. I.E., at 3 g the ramjet needs to operate for about 30 seconds to accelerate from 2,000 fps to 5,000 fps.
Exit velocity given an acceleration and tube length
v = sqrt(2as) ex a = 32.2 ft/sec s = 100 ft v = sqrt([2]x[32.2 ft/sec.sec]x[100 ft]) = Velocity (ft/sec) Table Distance (ft) a (g) 10 20 30 50 100 200 300 1 25 36 44 57 80 113 139 3 44 62 76 98 139 197 241 5 57 80 98 127 179 254 311 10 80 113 139 179 254 359 440 20 113 160 197 254 359 508 622 50 179 254 311 401 567 802 983 100 254 359 440 567 802 1,135 1,390 200 359 508 622 802 1,135 1,605 1,966
Travel Time for a Launch Vehicle in a Launch Tube
t = v/a ex v = 2,000 ft/sec a = 100 g (3220 fps) 2000 ft/sec t = -------------- = 0.62 sec 3220 ft/sec.sec Table of Tube Travel Times a (g) 400 800 1200 1600 2000 1 12.42 24.84 37.27 49.69 62.11 3 4.14 8.28 12.42 16.56 20.70 5 2.48 4.97 7.45 9.94 12.42 10 1.24 2.48 3.73 4.97 6.21 20 0.62 1.24 1.86 2.48 3.11 50 0.25 0.50 0.75 0.99 1.24 100 0.12 0.25 0.37 0.50 0.62 200 0.06 0.12 0.19 0.25 0.31 400 0.03 0.06 0.09 0.12 0.16 Notes: 1. This is used for determining the time it takes the launch vehicle to travel down the tube. With that known you can estimate how fast the gas valve must act. The valve should complete actuation in no more than one tenth the travel time. I.E., if the travel time is 0.3 seconds then the valve should complete actuation in 0.03 seconds.
Cross Sectional Area of a Tube
A = Pi.r.r or Pi.D.D/4 ex D = 6 in (r = 3 in) (3.414)x(6 in)x(6 in) A = --------------------- = 28.7 in.in 4 Table of Areas Tube Cross Sectional Diameter (in) Area (in.in) 1 0.8 2 3.1 4 12.6 6 28 8 50 10 78 12 113
Force exerted on the launch vehicle in a pressurized tube
f = pA = p.(Pi.r.r) ex p = 100 psi D = 6 in r = 3 in A = Pi.(3 in x 3 in) = 28.3 in.in f = (100 lbf/in.in)x(28.3 in.in) = 2,830 lbf Table of Forces Tube Cross Gas Force Diameter Sectional Pressure Created (inches) Area (in.in) (psi) (lbf) ------------------------------------------------ 1 0.78 50 39.2 1 0.78 100 78.5 1 0.78 200 157 4 12.6 100 1,257 6 28.3 100 2,830 12 113 100 11,310 Additional Table of Forces Tube Dia Gas Pressure (psi) (inches) 20 50 100 200 500 1 16 39 78 157 393 4 251 628 1,257 2,513 6,283 6 566 1,415 2,830 5,660 14,150 12 2.262 5,655 11,310 22,619 56,549 24 9,048 22,619 45,239 90,478 226,195 48 36,191 90,478 180,956 361,911 904,779 72 81,430 203,575 407,150 814,301 2,035,752 120 226,195 565,487 1,130,973 2,261,947 5,654,867
Achievable Amateur Rocketry Performance Using Tri-Mode Launch Assist
System Ramjet Cutoff Rocket LV Coast Description fps ft fps Altitude (ft) Ramjet to Mach 5, 60 Degree Angle, No Rocket Motor 5,000 100,000 0 391,000 Ramjet to Mach 5, 90 Degree Angle, No Rocket Motor 5,000 100,000 Ramjet to Mach 5, 60 Degree Angle, 1,100 fps Rocket 5,000 100,000 1,100 548,000 (104 mi) Ramjet to Mach 5, 90 Degree Angle, 1,100 fps Rocket 5,000 100,000 2,500 973,440 (184 mi) Ramjet to Mach 7, 60 Degree Angle, No Rocket Motor 7,000 120,000 Ramjet to Mach 7, 60 Degree Angle, 1,100 fps Rocket 7,000 120,000 1,100 881,000 (167 mi) Ramjet to Mach 7, 90 Degree Angle, No Rocket Motor 7,000 120,000 Ramjet to Mach 7, 90 Degree Angle, 2,500 fps Rocket 7,000 120,000 2,500 1,401,400 (265 mi)
This Page Last Updated 22 Dec 98