________________________________________ GURPS Vehicles 2nd Edition Additions MA Lloyd (malloy00@io.com) 9 August 1998 Modifications and Additions to Chapter 10: Performance ________________________________________ p128 Cassidy is exaggerating, at 11 mpg flat out, Kitty Hawk gets rotten gas mileage. p128 Ground Speed. A better formula is P(kW) = (Adr/4) x (mph/100)^3 + (Lwt/speed factor x10) x (mph/100). Unfortunately that's MUCH uglier solved for mph, the sum of two cube roots, one of which will be negative and both containing square roots under the cube root radicals.... This works fairly well for wheeled vehicles, but it is necessary to lower the speed factors for non-wheels by a factor of at least two to keep the rolling friction term reasonable. p129 Ground Acceleration. Use (21.9/Lwt) x [Thrust + (1000xkW/Top Speed)]. This produces similar results in the ground vehicle envelope and fixes the problem of thrusters with higher gAccel than sAccel. It is also unit consistent. p129 gMR. Multiply gMR by the local gravity. p129 gMR Modifiers. Subtract 0.25 (or halve gMR if 0.25 or less) if a wheeled vehicle lacks tires. p130 Hydrodynamic Drag. Compute hydrodynamic drag as (1.75/Hl) x surface area x the square root of [(loaded weight - contragravity lift)/flotation]. Multiply drag by the liquid density for liquids other than water. p131 Water Acceleration. The formula in Vehicles is the free space case, but water resistance does matter. Use 20 x [Ath - Hdr(Top Speed/10)^3]/Lwt instead. p132 wSR. If the vehicle isn't a catamaran or floating on pontoons, subtract 2 from wSR if Lwt is less than 1/4 the maximum flotation. p132 Submerged Hydrodynamic Drag. Use Hdr = 1.5 Surface Area/Ls. In liquids other than water, multiply by the liquid density (in g/cc) p133 Underwater Top Speed. Use Speed = 2.4 x square root of (Sth/Hdr) p133 Underwater Draft. More plausible results are obtained by dividing by 6 rather than 3, implying about a 5:1 length:draft. p133 Crush Depth. Divide crush depth by the surface gravity of the world. For liquids other than water also divide by the density of the liquid. p134 Hover Capability. p156 gives a reverse limit of 1/2 top speed or 150 mph, not the 1/20 given here. Compromise on 1/10 top speed, maximum 150 mph. p134 Flarecraft. Drop the 'maximum of 6 feet'. p134 Aerial Top Speed. Multiply drag by the density of the atmosphere. p134 Aerial Top Speed. D. Instead of 5 per hardpoint, use 2 x load surface area (computed from VPS etc.)/Sl (typically 20 for missiles) p135 Aerial Top Speed Add Ducted Fans 600mph to the maximum speed table. p135*Aerial Maneuver Rating, change the last sentence 'lower result' to 'higher result' p135 Aerial Maneuver Rating. An aircraft with vectored thrust has a minimum aMR of (vectored thrust/Lwt), though diverting thrust from aerostatic lift to aMR may cause the vehicle to fall. p136 Mag-Lev Performance. If the mag-lev runs in an evacuated tunnel, or in vacuum, drag is technically 0, but use surface area/40 instead. p136*Mag-Lev Performance. Deceleration in mph/s is 40 x mMR. ________________________________________ Performance p129a Ground Stability and Maneuver ________________________________________ A case can be made that ideal friction limits all gMRs to the local gravity or less unless the vehicle somehow bites into the ground. I liked Chris Dicely's solution: cut gMR to 0.5, give a +1 to control rolls for every 0.5 or fraction thereof the original gMR was reduced and replace the 2xMR limit on maneuvers (p151) with 1.0G, or 1.25G for wheeled vehicles with tires. In any case the gMR values for legs and flexibodies are extremely unrealistic; even proportionally, at minimum halve them. ________________________________________ Performance p132a Can the Vehicle Submerge? ________________________________________ To function as a submersible, the vehicle must be able to achieve neutral buoyancy, where its flotation (62.5 x its volume) is exactly balanced by its weight and lift. Most submersibles adjust buoyancy by filling or emptying ballast tanks. The default submersible hull can submerge if its loaded weight is between 0.8 and 1.0 x its flotation. The traditional method of adding weight to a submersible that is too light is to install tanks of mercury, but adding armor works too. Vehicles without ballast tanks can function as submersibles if they have some other method of adjusting effective weight or flotation to equality. Active flotation (p32a) or hydrodynamic lift for example. Hydrodynamic lift is the exact analog of aerodynamic lift, though it is more common in animals than in vehicles; for example most sharks sink if they stop moving. Install wings (usually called fins or planes on a water vehicle) or a lifting body hull, find the lift area as for aerial stall speed (p133) and compute the stall speed below which the vehicle sinks to the bottom (or bobs to the surface) equal to 0.25 x [(79+Hl)/80] x square root of (needed lift/lift area). ________________________________________ Performance p133a Underwater Driving ________________________________________ A ground vehicle too dense to float (>62.5 lb/cf) and able to function underwater may be able to drive along the bottom. Most underwater terrain counts as Quagmire (p153) though there are occasional exceptions. Underwater drag is much higher than in air, divide speed and gAccel by 9 and add 20mph/s to gDecel to account for it. When computing ground pressure underwater, flotation subtracts from vehicle weight just as contragravity does in air. ________________________________________ Performance p133a Stall Speed ________________________________________ Properly stall speed is 7 x Sl x Rs x square root of [(Lwt-Static Lift)/Lift Area]. Multiply it by the square root of the local gravity and divide by the square root of local air density. Stall speed is the speed an aircraft stalls in free flight, but near the ground airfoils get the same lift enhancement as hoverfans. An aircraft landing or taking off from terrain passable to hovercraft stalls at 0.71 x Stall Speed. This reduces takeoff speed, although unless air speed exceeds full stall speed the plane can't lift more than a few feet into the air. Seaplanes are even better off, as they start to lift out of the water drag drops, increasing water speed. If water speed exceeds 0.61 x stall speed, increase it to 1.0 x stall speed. Do not apply these modifiers to flarecraft, which already get an area multiplier to reflect the effect. _____________________________ Performance p136a Glider Performance _____________________________ The important statistics for a glider are its minimum and maximum forward speeds, terminal velocity, and glide ratio. Minimum speed is simply stall speed = 7 Rs Sl square root of (Lwt/lift area). Terminal velocity is the square root of (7500 Lwt/Adr). Maximum forward speed equal to 0.4 x terminal velocity. Glide ratio is (stall speed/maximum speed)^2 A glider is always sinking at its glide ratio times its current forward speed, but speeds are relative to the air, not the ground, so tailwinds (which add to ground speed without increasing air speed) and updrafts (which subtract from the sinking rate relative to the ground) are very important. For comparison 2 mph updrafts are easily located, 5 mph updrafts are fairly common in thermals or above ridges or other places moving air is deflected up by ground irregularities, and updrafts over 20 mph are probably not available outside of a storm front. ________________________________________ Performance p136a Space Performance ________________________________________ *Specific Impulse* Specific Impulse (Isp) is the standard measure of rocket performance. It is actually an effective exhaust velocity, but for historical reasons connected to the two kinds of pounds (mass and force) in the US it is normally quoted in units of 'seconds'. To find the specific impulse of an engine, divide its thrust by the amount of fuel it uses per unit time. To obtain Isp in seconds: Isp(sec) = 3600(sec/hr)/[Fuel Usage (gal/hr per lb) * Fuel Weight (lb/gal)] Specific impulse is given on the Reaction Engine Table above, for the engines in Vehicles it works out to: 6 Liquid Fuel Rocket 240 7 Ion Drive 29400 7 Liquid Fuel Rocket 290 7 Fission 860 8 Liquid Fuel Rocket 330 8 Fission 4320 8 MOX Rocket 280 7 Orion < 200,000 3 Solid Rocket 32 9 Fusion Rocket 21,600 4 Solid Rocket 70 9 Optimized Fusion 1,550,000 5 Solid Rocket 200 9 Antimatter Thermal 8,600 6 Solid Rocket 260 10 Antimatter Pion 20,500,000 (corrected) 7 Solid Rocket 335 8 Solid Rocket 390 Many other statistics can be computed from Isp, such as: Effective exhaust velocity = 1 gravity * Isp(sec) Jet Power (kW) = 0.0219 * Isp(sec) * thrust (lb) The most important is delta-v, the total velocity change a spacecraft can make given a particular amount of fuel. This is computed from the Rocket Equation: delV = Exhaust velocity * ln (Mass Ratio) delV(mph) = 21.9 * Isp(sec) * ln (1 + [fuel weight(lbs)/dry wt(lbs)]) Alternately, if you know the delta-v needed, you can compute how much fuel will be required. First compute: k = e ^ [-delV(mph)/(21.9 * Isp(sec))], which will be a number from 0 to 1 If you are willing to end the burn with dry tanks the fuel needed is fuel (lb) = dry weight (lb) * [(1-k)/k] If you start with a known amount of fuel, the amount left is fuel left (lb) = [k * initial fuel (lb)] - [(1-k) * dry wt (lb)] If fuel left comes out negative, you don't have enough fuel to make the desired velocity change! To determine the duration of a burn, either compute the time it takes the engine to burn that much fuel (constant thrust) or how long it takes to accelerate to that speed at the initial sAccel of the ship (constant g) If you are using these formulas in the design process to compute how big a fuel tank you need for a particular mission, don't forget the empty tank adds to the dry weight. I have gotten helpful first estimates by adding twice the tank weight (in lb/gal) to the fuel weight (in lb/gal) and using that to compute Isp, but you will probably need to iterate your solutions. *Multiple Stages* Space vehicles are often built with multiple stages. Compute the performance of the last stage alone. Then build the next stage treating the prior stage as a component and so on. The advantage of multiple stages is the independently computed delta-vs add directly, providing more total delta-v that if all the empty lower stages were boosted to the full velocity. *Density and Specific Impulse* The highest specific impulse fuel does not always give the best total performance, sometimes denser fuels are better. This is particularly true of aircraft, where less dense fuels mean larger tanks, hence more surface area and more drag, but it's true any time fuel is a large fraction of the vehicle loaded weight. Consider a TL7 rocket that is 90% HO fuel (2.1 lb/gal, 460 seconds Isp). It has a total delta-v of g x Isp x ln (1 + fuel wt/dry wt) = 21.9 x 460 x ln (1 + 90/10) = 23200 mph. Now drain the fuel and replace it with kerosene/LOX (8.5 lb/gal, 346 sec), a lower specific impulse but 4 times heavier. Dry weight is unchanged, so total delta-v is now 21.9 x 346 x ln (1 + 4.0*90/10) = 27400 mph, not a trivial difference.