How the Motor Matchup Drag Race Simulator Works

The basic idea
Speed and acceleration
Engine, gears and wheels
Tires and traction
Shifting
Aerodynamics
Fans and thrusters
Vehicle data
FAQ
Further reading

The basic idea

The simulation is built on Newton's second law. Add up the forces on the car, divide by its mass, and you get its acceleration at that moment.

The forces depend on each other. Engine output depends on rpm, rpm depends on wheel speed and gear, grip depends on how much weight is on the driven tires, and drag grows with speed. So the simulator works in small steps. Each step it calculates every force, advances time by a hundredth of a second, and repeats. A quarter mile takes around 1,200 steps.

Add up the forces
Engine thrust through the gears and tires, minus air drag and rolling resistance.
$F_{net} = F_{tires} - F_{drag} - F_{roll}$
Divide by mass
Curb weight plus a driver, plus the rotating parts that also have to be sped up.
$a = \frac{F_{net}}{m}$
Step forward in time
Update speed from the acceleration, then position from the speed.
$v += a Δt, x += v Δt$
Update the machine
New rpm from wheel speed, shift if needed, update weight transfer and tire slip. Repeat.
$Δt = 10 ms$

The math

The integration method is semi-implicit Euler: speed is updated from the new acceleration first, then position from the new speed. It stays stable over many small steps. Some parts of a run use smaller steps, such as a launch through a torque converter, and the tire model takes extra sub-steps when a wheel is spinning hard.

m a = F_{tires} - \frac{1}{2} ρ C_{d} A v^{2} - C_{rr} m g

m: mass: curb weight plus a driver, plus the rotating inertia of wheels and drivetrain
ρ: air density, from altitude and temperature
Cd · A: drag coefficient times frontal area (section 05)
Crr: rolling resistance coefficient of the tire

Speed and acceleration

These two charts show the selected race. Acceleration is the slope of the speed line, so it shows where a car gains and loses time.

Acceleration is highest near the start and drops as speed rises, because the gears get taller and drag increases. The dips in a gas car's acceleration line are gear changes. A single-speed electric car has no dips; its acceleration falls off as the motor reaches its power limit.

Speed

2022 Model 30 mph2024 M40 mph

Acceleration

2022 Model 30.30 g2024 M40.58 g

1 g is the acceleration of a falling object (about 22 mph per second). Street tires on dry pavement can only support a little more than that.

Result	0-60 mph	60 ft	¼ mile	Trap
2022 Model 3	3.48 s	1.92 s	11.81 s	116.3 mph
2024 M4	3.85 s	2.05 s	11.86 s	122.6 mph

How the times are measured

Times follow drag strip and magazine practice. The clock starts after one foot of rollout, the distance a car moves before it leaves the starting beam at a real strip, so a published 0-60 is a little quicker than one timed from a dead stop. The race clock above runs from the green light instead, so it reads a few tenths more than the quarter-mile time. The 60-foot time mostly reflects the launch. Trap speed is measured over the last stretch before the finish and mostly reflects power.

Engine, gears and wheels

The engine produces torque. The gears multiply it, and the tires turn it into a force that pushes the car forward.

Each car has a torque curve, the torque the engine or motor makes at each rpm. Horsepower is calculated from torque and rpm, which is why the two curves cross at 5,252 rpm. Electric motors make full torque from zero rpm, then torque falls off once they reach their power limit.

Torque and power

torque (lb-ft)horsepower

2022 Model 3

2024 M4

The dot shows each car's rpm and torque at the current point in the race.

Lower gears multiply torque more, so first gear can often produce more force than the tires can use. Higher gears produce less force but reach higher speeds. The chart below plots the force at the tires in each gear against road speed. Engineers call this a tractive effort diagram.

The solid line is the force the car actually put down in the race. When it follows the dashed grip limit, the car is limited by traction. When it follows a gear curve, it is limited by the engine or motor. The sharp dips are shifts. The dark line at the bottom is drag plus rolling resistance. The car keeps accelerating as long as the force at the tires is above that line.

Force at the tires, by gear

deliveredeach gear at full throttlegrip limitroad load

2022 Model 3 · Single-speed

2024 M4 · 8-speed automatic

Both panels share the same scales. Gear curves assume the engine is locked to the wheels. Near a standstill the delivered line can sit above first gear: the clutch or torque converter slips so the engine can stay up at launch rpm, and a converter multiplies torque as well.

The math

Force at the tires is engine torque multiplied through the gears, less driveline losses, divided by the tire radius:

F = \frac{τ \cdot i_{gear} \cdot i_{final} \cdot η}{r}

When the clutch is engaged, the same ratios tie engine speed to road speed. Power is torque times rpm:

rpm = \frac{v \cdot i_{gear} \cdot i_{final}}{2 π r} \cdot 60 hp = \frac{τ \cdot rpm}{5252}

τ: engine or motor torque at the current rpm (lb-ft in the charts, N·m inside the sim)
i: gear ratio and final drive ratio
η: driveline efficiency: what survives the gearbox, differential and, on AWD cars, the transfer case
r: tire rolling radius, which grows slightly with speed and shrinks under load

Torque curves are built from published peak torque and power and the rpm of each peak, or taken from a measured dyno curve where one exists. Turbo lag is modeled. Driveline efficiency depends on the type of gearbox, the final drive and whether the car is all-wheel drive.

Tires and traction

A tire produces the most grip when it is slipping slightly. Past that point grip drops, which is why too much wheelspin slows a launch.

A driven tire always turns a little faster than the car is moving. The difference is called the slip ratio. Grip rises quickly over the first few percent of slip, peaks, then falls off as the tire spins. Each car in the simulator has a curve like this, based on its tire compound, tire width, the load on the tire and temperature.

Grip versus slip

2022 Model 32024 M4

A typical road-tire shape, drawn with Pacejka's magic formula; real curves vary by tire, load and temperature. The dots are each car's live slip from the race.

Grip also depends on how much weight is on the tire. Under acceleration, weight shifts from the front axle to the rear. That helps rear-drive cars and hurts front-drive cars. All-wheel-drive cars can send torque to whichever axle has more grip.

Traction at launch

deliveredgrip limit (μ × load)

2022 Model 3 · AWD

2024 M4 · RWD

Where the two lines meet, the car is traction limited. Where the solid line drops below the dashed line, the engine or motor is the limit.

Driven tire slip

2022 Model 32024 M4

Weight on the front axle

2022 Model 349%2024 M451%

The math

s = \frac{ω r - v}{v} F_{x} \leq μ (s) \cdot F_{z} Δ F_{z} = \frac{m a h}{L}

ω r: surface speed of the driven tire
μ(s): friction coefficient at that slip: the curve above, scaled to the tire
Fz: load pressing the tire into the road
h, L: center of gravity height and wheelbase

Each driven axle is simulated separately. Engine torque speeds the wheel up, the road pushes back through the tire, and wheel speed is updated every step. Grip does not scale exactly with load: doubling the weight on a tire gives somewhat less than double the grip. Wider tires and stickier compounds have more grip, and grip changes with temperature.

Launch control and traction control reduce torque when slip gets too high, aiming to keep the tire near the peak of its curve. Weight transfer builds up over a short time rather than instantly.

Shifting

A gas engine only makes useful power over part of its rpm range, so it needs several gears to cover a quarter mile.

The simulator upshifts at redline, or earlier if the next gear would already give more force at the wheels. Some drive is lost during each shift, and how much depends on the type of transmission.

Drive torque through one upshift

Manual

Drive is cut completely while the clutch is in and the engine drops to the new gear's rpm.

Automatic

Clutches inside the gearbox hand off between gears. Torque drops but not to zero.

Dual-clutch

The next gear is preselected on a second clutch, so the handoff takes a few hundredths of a second.

Simplified shapes for illustration. Shift times come from the manufacturer where published, and are otherwise estimated from the transmission type and age.

Engine and motor speed

2022 Model 3gear 12024 M4gear 1

Each drop is an upshift. A single-speed electric car has no shifts, so its motor speed rises the whole way.

Aerodynamics

Drag increases with the square of speed, and the power needed to overcome it increases with the cube. Doubling speed means four times the drag and eight times the power.

Drag is negligible at the start of a race. By the end of a quarter mile it is usually the largest force working against the car, and it is what limits top speed. A car's drag depends on two numbers: its drag coefficient (Cd), which describes the shape, and its frontal area.

2022 Model 3

C_d 0.23 · A 2.27 m² (est)

2024 M4

C_d 0.34 · A 2.23 m² (est)

Resistance at speed

2022 Model 30 lbf2024 M40 lbfrolling resistance

Solid lines are aerodynamic drag. Faint lines are rolling resistance from the tires, which changes little with speed.

Power needed to hold a speed

2022 Model 30 hp2024 M40 hp

The horizontal lines are each car's rated peak power. The gap between that line and the curve, less driveline losses, is the power left over for accelerating.

The math

F_{d} = \frac{1}{2} ρ C_{d} A v^{2} P = F \cdot v \propto v^{3}

ρ: air density: about 1.22 kg/m³ at sea level and 60 °F, less at altitude or on a hot day
Cd: drag coefficient, from the manufacturer where published
A: frontal area: published, or estimated from the car's width and height

Air density is calculated from altitude and temperature. Denser air means more drag, and gas engines lose power at altitude. Cars with published downforce figures include it: downforce adds load on the tires, which adds grip.

Fans and thrusters

Every other car on this page speeds up one way: the tires push on the road. The 2028 Tesla Roadster adds two more. Fans suck it down onto the road so the tires can push harder, and nitrogen thrusters push it forward without touching the road at all.

The Base trim has neither, the Performance adds the fans, and the Thrust adds the thrusters. Every chart below is a real run of each car through the simulator.

Base

0-60 2.8 s1/4 mi 9.5 s

Performance

0-60 1.5 s1/4 mi 8.0 s

Thrust

0-60 0.8 s1/4 mi 7.2 s

The forces at the launch

Weight and the fans press the car into the road; the tires and the jet push it forward. Drag only builds once it moves.

Performance against Thrust

Performance

0mph0.29g-kN jet-psi

Thrust

0mph0.79g9.5kN jet9,967psi

0.00 s

Same body, same fans, same motors. The difference is the jet.

Fans: grip before the car moves

Wings make downforce from moving air, so they do nothing at a standstill, exactly when a drag car needs grip most. Fans pull air out from under the car at any speed. In our model they pull down about as hard as the car weighs, so from the first inch the tires can push nearly twice as hard.

Load on the tires

weight+ fans+ body downforce7,620 lbf

Roadster Performance. The fans add the same load at every speed; the wing and underbody add more as speed rises.

Acceleration in the first three seconds

BasePerformance (fans)0.29Thrust (fans + jet)0.79

All three are limited by traction at the launch: the fans raise that limit, the jet adds force the tires never see.

Thrusters: a push that skips the tires

The tank holds about 100 kg of nitrogen at 10,000 psi. Open the valves and it rushes out of the nozzles faster than sound, pushing the car forward like a rocket, with no combustion. The push is biggest in the first instant and fades as the tank empties.

The jet

thrust, all nozzles9.5 kN

Roadster Thrust, launch to the finish. The valves take 50 ms to open, so the peak comes just after the launch.

Inside the tank

Expanding gas gets very cold. With about a fifth of the nitrogen left it starts to condense into liquid and the tank boils, which keeps a weak jet going for seconds.

Pressure and temperature

gas left99.9 kgtank60 °F

Pressure

Temperature

When to fire

In a race you choose the speed the thrusters fire at. The gas gives about the same push whenever it goes, but it saves the most time at the launch, where the tires are at their limit. Firing late trades time for trap speed. Fired at the launch the Thrust runs about 7.2 s; fired at 150 mph, about 8.2 s but some 26 mph faster through the traps.

Quarter mile by fire point

ThrustPerformance (no jet)

Elapsed time

Trap speed

Each point is a separate run with the thrusters set to fire at that speed.

The math

Newton's second law with the jet added. The car gets lighter as the gas leaves:

m (t) a = F_{tires} + F_{jet} - F_{d} - F_{rr}

The most the tires can push grows with the load on them, and the fans add load at any speed:

F_{tires} \leq μ (m g + F_{fans} + F_{body} (v))

Thrust is the momentum of the gas thrown backward, plus the exit pressure over the outside air's:

F_{jet} = n (\dot{m} v_{e} + (p_{e} - p_{a}) A_{e})

The flow through each nozzle's narrowest point runs at the speed of sound, so the mass flow, and with it the thrust, follows the tank:

\dot{m} = ρ^{*} a^{*} A_{t}

Gas that expands without time to take in heat cools along its isentrope. For an ideal gas:

\frac{T}{T_{0}} = {(\frac{p}{p_{0}})}^{(γ - 1) / γ}

μ: tire friction coefficient: how hard a tire can push per pound pressing it down
F fans: the fans' pull, the same at every speed while the battery feeds them
F body: downforce from the wing and underbody, growing with speed squared
n: nozzle count (4 rear on the Thrust)
ṁ, ve: each nozzle's mass flow and exit velocity
pe, pa, Ae: exit pressure, outside air pressure and nozzle exit area
ρ*, a*, At: gas density and sound speed at the throat, and the throat's area
γ: ratio of specific heats, 1.4 for nitrogen

Nitrogen at 10,000 psi is far from an ideal gas, so the simulator reads its state from reference tables for real nitrogen, which cools faster than the formula says. Once it reaches its dew point the tank boils, and an energy balance on the boiling liquid and vapor sets its pressure.

Tesla's figures and our estimates

From Tesla (as reported)

200 L carbon tank, up to 10,000 psi of nitrogen
Four rear thrusters, over 80 kN together
Valves shut after about five seconds
Thrust trim: 0-60 under 0.9 s claimed
Fans: over 2 g of cornering from 0 to 50 mph

Our estimates

Fan pull equal to the car's weight
About 95 kN from the four nozzles on a full tank
Curb weights, and the tank's and hardware's mass
Nozzle shape, valve timing and where the tank sits
Drag coefficient and gearing

Tesla has not published full specifications. We will update the cars when it does.

Race the Performance against the Thrust

Vehicle data

Every car starts from published specifications, collected by hand from manufacturers and other reliable sources.

Peak power and torque, gear ratios, final drive, curb weight, tire sizes and dimensions are usually published. Drag coefficient, frontal area, shift time, weight distribution, center of gravity height and the shape of the torque curve often are not. When a value is missing, we estimate it from the car's type, age and similar cars in the catalog.

Some manufacturers underrate their engines. Where independent dyno results show a car makes more than its rated power, the simulator adjusts for it.

We compare results against published instrumented tests. If you see a number that looks wrong, you can report it on the data page.

FAQ

Why don't your times match the magazine test I read?

Real-world times depend on the driver, surface, weather and the individual car, and different tests of the same model often disagree. The simulator runs every car in the same conditions (dry pavement, sea level, 60 °F, no wind) so cars can be compared directly.

Can you add a car?

If its key specifications are available, yes. Start here.

Do modifications use the same physics?

Yes. Modifications like more power, different tires or less weight change the car's inputs, and the race is simulated again with the new values.

Is downforce included?

Yes, for cars with published downforce figures. For most road cars it is small at quarter-mile speeds and isn't published, so it is not included.

Data glossary

The specifications the simulator uses.

Engine and motors

Horsepower, torque: Peak output, and the rpm where each peak occurs
Torque curve: Torque across the rpm range, published or built from the peaks
Redline: Highest engine or motor speed in normal use
Induction: Naturally aspirated, turbocharged or supercharged
Layout: Cylinder count and arrangement, displacement, bore and stroke, valvetrain, compression ratio
Battery capacity: Usable energy in kWh for hybrids and EVs

Transmission and drivetrain

Transmission: Manual, automatic, dual-clutch, CVT or single-speed, and the number of gears
Gear ratios: Ratio of every gear
Final drive: Axle ratio, the last reduction before the wheels
Driven wheels: Front, rear or all-wheel drive, and how AWD splits torque
Shift time: How long drive is interrupted during an upshift (often estimated)

Body and tires

Curb weight: Weight ready to drive, without occupants. A driver is added for every race
Weight distribution: Share of weight on each axle at rest
Dimensions: Length, width, height, wheelbase, track and ground clearance
Drag coefficient, frontal area: Aerodynamic slipperiness and size (often estimated)
Tires: Size front and rear, and compound, from all-season to drag radial

Contact

Questions and corrections: brian@motormatchup.com.

Some 3D race cars are modified from the following works, licensed under CC BY 4.0: Generic passenger car pack by comrade1280; Top Fuel Dragster by rhinoart.

The basic idea

Speed and acceleration

Speed

Acceleration

Engine, gears and wheels

Torque and power

Force at the tires, by gear

Tires and traction

Grip versus slip

Traction at launch

Driven tire slip

Weight on the front axle

Shifting

Drive torque through one upshift

Engine and motor speed

Aerodynamics

Resistance at speed

Power needed to hold a speed

Fans and thrusters

The forces at the launch

Performance against Thrust

Fans: grip before the car moves

Load on the tires

Acceleration in the first three seconds

Thrusters: a push that skips the tires

The jet

Pressure and temperature

Quarter mile by fire point

From Tesla (as reported)

Our estimates

Vehicle data

FAQ

Why don't your times match the magazine test I read?

Can you add a car?

Do modifications use the same physics?

Is downforce included?

Further reading

Data glossary

Contact

Specs

Dragstrip

EV Efficiency

Mods

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Specs

Dragstrip

EV Efficiency

Mods

Beat This Car

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