Water Rockets - 2 Litre Dart Rocket
For people who are curious about what heppens when you propel a sixty gramme dart upwards with an acceleration of around 2 kilometers per second per second - 0 to 60mph in 26ms.

Water rockets, requiring a largish capacity for air and water, are usually large in diameter, this causing a large amount of drag and limiting the height acheived. However, the impulse rating for even a 2 litre water rocket is normally E - four times the impulse of a pyro motor that can be bought over the counter in a high street toy shop.

A dart is one way of combining the impulse of a water rocket with a small diameter rocket. The dart is effectively a passive sustainer of a 2 stage rocket, relying on the kinetic energy that the booster has given it during the first stage of ascent.

Kinetic energy is dependent upon mass and velocity and the booster is designed to give as much velocity to the dart as possible. The booster is optimised for the dart's height (which translates to initial velocity) and the simulation is therefore interactive - the performance of the booster and the dart affecting each other - and therefore, I have updated the simulator (dart included in version 4.30) to work with darts and optimise a 2 stage dart configuration (the optimisation being a divergent one if a small or zero thrust is used for the sustainer).

Materials

The dart has to be capable of withstanding an acceleration of between 100 and 210G (yes, that is 2 kilometers per second per second, making the 60g dart put a 12.5kg force on the launcher) so it needs to be fairly strong. I opted to make it from 21.5mm white uPVC overflow pipe. The target weight for the dart is between 60 and 80g so this was fairly easy. The materials list is as follows . . .

Dart

  • 25cm of 21.5mm outside diameter uPVC overflow pipe;
  • A cork;
  • Some flat PET bottle side;
  • Red paint;
  • A candle (or some epoxy resin and lead shot (possibly a small radio transmitter) - see below); and,
  • Some paper.

Connector

  • A PET bottle;
  • The top of a PET bottle;
  • 8cm of 21.5mm outside diameter uPVC overflow pipe with some 1cm long rings made from it as well;
  • A few cm of pipe that will fit inside (19mm o/d pipe - I got some from a Halloween toy belonging to the kids'); and,
  • 20cm of 15mm PVC Pipe (this stuff is marked with the following - "Speedfit 15mm B-PEX to BS7291 : Part 3:1990 Class S 12 BAR 20ºC - 4 BAR 82ºC - 3 BAR 92ºC 9927/0609/279". It just fits into the gap down the middle and will hold the dart in place).

Tools et cetera

  • Super glue (be careful with this stuff - it is adults only, ie, get an adult to do it for you (this may also apply to adults) and read the instructions on the pack);
  • Sand paper;
  • Coarse file, Surfform or a grater (for trimming cork so a kitchen grater is good enough - it is what I used);
  • Masking tape (or Sellotape);
  • Pair of scissors;
  • Pipe Cutter; and,
  • Sharp(ish) knife.
Diagram of construction of Dart Rocket.
 
A Candle Wax
B PET Sheet Fins
C PET Bottle Parts
D 19mm tubing
E Cork
F Cut-down uPVC
support
G 21.5mm uPVC tubing
of Dart
H 21.5mm uPVC tubing
of Connector
I Hi-Tech cellulose
and polymer
composite material
(see method)
J 15mm pipe
 
1 Connector
2 Dart
3 Top of Bottle
 

Construction

Dart

First, cut around 25cm of the 21.5mm o/d uPVC pipe using the pipe cutter and then clean the end burrs using the knife. Next, push around 1cm of the cork into the end of the pipe you have just cut and then, using the knife, cut the end of the cork to form a rough, elliptical form, using the pipe as a handle. Try to get roughly the right shape and finish up with the file or grater. Once you have got it right glue it in place with super glue - this is where the fun starts.

You are now going to make a high technology cellulose and polymer composite material to cover the nose cone (this is rocket science). Tear strips of paper, approximately 5mm by 3cm (roughly 0.25" by 1.5") such that all four edges of the strip are torn - that is to say that the transition from full thickness to edge is gradual over around 1mm. Put some super glue on the strip and stick it onto the cork, wrapping it around the exposed cork and pressing it into place.

NOTE, I used my thumb to do this and I found that if I kept it moving on the surface, my thumb did not stick to it - when you come to the end of each smoothing action, lift your thumb before it stops on the surface, otherwise it will stick to it.

Dab glue on the edges where the paper has not stuck down to the cork, pressing down and keeping your finger moving. Repeat this process until all of the cork is covered with paper and glue. Then, hold the sand paper in one hand and roll the dart along your thigh (as you sit), sanding the corners off the plastic/paper composite nose cone (imagine you were rolling a Cuban cigar - not that I ever have). An alternative to this is to put it in a lathe (but I haven't got one).

Repeat this process - adding a layer of paper and then sanding - until a smooth nose cone is formed. It may start off a bit knobbly but after a while, you will get a smooth elliptical nose cone to be proud of. After your final sanding, apply one further layer of superglue to make a nice smooth finish. You should now have a smooth, well shaped nose cone on your dart and a thick (0.3mm (ish)) layer of superglue on your thumb. Superglue this thick comes off (relatively) easily so a soak in some warm water and some teasing over a few minutes with a spoon handle should get rid of almost all of it (like dealing with superglue at any other time - the rest will wear off in a few days ;-).


Cut the fins out to this shape.
Right-Click and Zoom In to see betterOnce your thumb is clean (or while your assistant is cleaning his or her thumb), you can cut out the fins (as per the sketch on the right - around 10cm long) with a pair of scissors, fold the PET on the dashed lines and then make a smooth curve the other way between the dashed lines - in this way, it follows the curve of the uPVC tubing and the fins stick out at 90 degrees to the dart body. Carefully glue them into place with superglue (as per the sketch on the left). End view of Fins, glued into place
Right-Click and Zoom In to see betterNote that the dart does not hold any pressure so superglue is okay on it.

After this, put some masking tape around the dart body, just behind the nose cone and paint the nose cone red, pealing off the masking tape before it dries so that you end up with a smooth transition again.

Next, weigh the dart and then set it upright so that it is pointing down and can be left for a while. Light a candle and drip molten wax into the dart, weighing it occasionally, until it weighs enough. Let it set until the dart is cold to the touch and then drop some super glue down the inside (10 to 20 drops so that it bonds with the uPVC pipe and forms a layer or ledge that will stop the wax from falling out on acceleration.

Wax is cheap, availble world-wide and dense enough for this purpose. If you want to concentrate the extra weight towards the nose cone by using lead shot then you will need something that is stronger than wax. Epoxy resing should do the job well enough. Your dart is now ready.

Connector

The connector has to be able to transmit over 12 kilogrammes force from the top of the rocket pressure vessel to the bottom of the dart. Fortunately, the top of the pressure vessel in the centre is pretty strong and by using a piece of pipe the same diameter as the dart, all of the force is transmitted directly. An additional requirement is that the dart should separate easily, as soon as the booster starts to slow down. With this in mind, I came up with the following . . .

I cut off the top of a 2 litre PET bottle so that it could slide over the bottom of another 2 litre bottle, forming a neat holder. I cut roughly 8cm of 21.5mm uPVC using the pipe cutter and deburred the ends using the sharp knife. This pipe fits snuggly into the top of the 2 litre PET bottle, the nozzle of which holds the pipe straight - being essentially cylindrical, stopping the pipe from rotating away from the axis of the bottle.

Then, I cut roughly 1cm of the 19mm diameter pipe and slid that into the end of the uPVC pipe so that around 1.5mm protruded, forming a ledge that would stop the dart from slipping out of its location. This was then glued into place and sanded slightly so that the dart was a loose fit. Then I cut two 1cm lengths of white uPVC pipe and one of the 19mm Halloween pipe, cutting a piece out of the side of each so that they could form a narrower cylinder.

I then rolled these up, putting a uPVC ring inside the piece of Halloween pipe that forms a ledge and a piece of each at the other end of the pipe so that they were flush with the bottom of the pipe. I then cut a 20cm length of the 15mm pipe, deburring it with the knife and sanding one end. Next, I pushed this through the end with the ledge and to the other end, until it was also flush.

This now allows the dart to sit on top, with the 15mm pipe forming a support that holds the dart in place around half way up and the black pipe at the bottom so that it doesn't slide off sideways. The connector slides through the nozzle of the top of the 2 litre PET bottle so that it can be taped in place and the connector slid through, making contact with the end of the pressure vessel as in the diagram above. Note that the uPVC is not glued into the neck of the bottle - this is because as the force on the connector increases during the launch, it could cause the bottle to buckle if it was glued. By allowing it to move inside the neck, the bottle only puts a sideways, straigtening force on the connector.

With the connector in the nozzle of the top of the bottle that I had cut, I positioned the the top of the bottle so that the rocket was straight with the sides (and, as this is an unmodified bottle, therefore with the nozzle as well) and then taped it into position. The booster rocket does not need fins because there is no opportunity for the dart to experience any significant effect - it is doing 60mph after just over 1/40th of a second.

Water

Using a 2 litre bottle (with connector + bottle weighing in at a total of 90g) and a 60g dart, the computer model gives the following predictions (velocities have been rounded to the nearest 5) . . .

Pressure Water Maximum
Acceleration
60g Dart
Weight
Maximum
Velocity
Height Time to
Apogee
Flight
Time
Impact
Velocity
psi g G kg m/s mph feet s s m/s mph
50 550 70 4.1 35 75 180 3.2 6.6 30 70
75 550 110 6.5 45 105 290 4.0 8.5 35 80
100 535 150 9.0 60 130 400 4.7 10.0 40 95
120 530 180 10.8 65 150 490 5.1 11.0 45 100
140 530 210 12.5 75 165 570 5.4 11.8 45 105

I have included in the table, the weight of the dart under the maximum acceleration of the rocket to give an idea of the forces involved and how strong the connector between the booster and the dart needs to be.

This dart has no recovery system and it is capable of travelling over 500 feet into the air, therefore great care must be taken to make sure that the rocket is launched vertically. The simulator shows that at an elevation of 42 degrees, the dart has a maximum range of around 280m (over 300 yards) at 140 psi, using the same amount of water; and a range of nearly 100m (109 yards) at 80 degrees elevation under the same conditions. It hits the ground at around 46m/s (105mph), therefore, make sure that everyone who may be affected by the launch knows what to expect. Launching onto a hard surface will obviously cause damage to the rocket so choose a soft surface (such as a soggy football pitch). In addition to this, it is in the air for a long time and therefore should be launched in calm/still conditions so that the air that the dart is launched into is still on-site by the time it lands.

When I tested it at 100 psi, the 60g dart landed in the side of a grassy hill and it went into the ground to a depth of 6" (150mm - the weather had been very wet). Knowing that the velocity of the dart on impact was around 42.7m/s and using equations derived from . . .

v = u + at and s = ut + 1/2at2
where (in this case):
v  =  initial velocity;
u  =  final velocity (which equals zero - normally, u and v would be assigned the other way around);
a  =  the value of the deceleration;
t  =  the duration of the deceleration; and,
s  =  the distance covered by the deceleration.

it is possible to calculate the value of the deceleration and the time that it took (assuming that the deceleration was smooth).

t  =  2s  =  0.15 x 2  =  7.0 ms   (hence the short thud, almost popping sound that I heard)
 
  v 42.7
 
a  =  v2  =  42.72  =  6.1kms-2   (or 620 G)
 
  2s 0.15 x 2

At 140 psi, the deceleration works out to be around 740G if the stopping distance is assumed to be the same. Finding the dart in the grass was difficult, so a radio transmitter onboard would have been a good idea except that anything that the dart carries has to be able to withstand accelerations of 740G and beyond. It goes without saying that landing on a harder surface would give a higher value for the deceleration.

The calculations reveal a surprise which is as follows:-

The force on the decelerating dart equals its mass multiplied by its deceleration ( f = ma ) so
  f = 0.060 x 6100 = 366 Newtons.
From this, the energy dicipated into the ground and the surrounding air during the deceleration (the Kinetic Energy of the dart on impact) equals the force multiplied by the distance ( KE = fs )
  KE = 366 x 0.15 = 55 Joules (you can get this from KE = 1/2mv2 and it gives the same answer but it is fun finding another way to arrive at it)
From this, the power output of the impact equals the energy divided by the time ( P = KE/t )
  P = 55 / 0.0070 = 7,860 Watts - equivalent to a fairly big PA system (albeit for only a short time) which is why I could here it from such a long distance away.

To the pictures . . .


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