arla-phy.htm Tri-Mode ARLA

Amateur Rocket Launch Assist (ARLA)

Physics of Motion


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)
Large Version (89k)

This Page Last Updated 22 Dec 98