  # Thrust versus Horsepower

### Thrust and Newton’s 2nd and 3rd Laws

So many people asking how to convert thrust to horsepower or vice versa. Whilst a seemingly simple enough question, the laws of physics and the power of jet engines all follow Netwon’s 2nd and 3rd laws. The question is answered almost unanimously, ‘you cannot’. Well what really gives with this – surely a Boeing 747 engine delivering 65 000 pounds of force could be converted to HP by a simple formula?  Well in simple terms as everyone wants to keep it simple, can you calculate the thrust at the wheels of a car?  Our minds can wrap around that one – just makes no sense. However, over the years its possibly one of the most covered questions in forums, magazines and journals and although to an aviation expert, seemingly mindless, the fact remains that most of us would like to just know how much horsepower those four Rolls-Royce RB211s are ‘putting out’. Time to recap…

If you do some research you’ll quickly find that the answers to this question vary from the probable to the ridiculous. To analyse properly one needs to look at these two formulae.

Horsepower and torque:

One HP is universally defined as the power required to lift 33,000 pounds one foot in one minute. Work is measured in foot-pounds, torque in pounds-foot.

HP = (Torque x RPM)/5252.  Torque = (HP/RPM) x 5252.

1 HP = 0.746kW

Aristotle, still a brilliant man of course, worked on the theory where there is velocity there is a force acting upon it, Newton declared that a force acting on it would create acceleration, the velocity would stay the same whether it had a force acting on it or not.

Thrust is a reaction force as described by Newton’s 2nd law, the vector sum of forces = mass x acceleration vector and 3rd law, for every action there is an equal and opposite reaction. The statement “fall-down drunk” is a good explanation of Newton’s third law.

For a rocket: Thrust = velocity exhaust gases x (change in mass / change in time) [mass flow rate]The above formula forms the very basis of rocket propulsion, velocity times change in mass over change in time. Horsepower is the rate at which work is done, thrust is a reaction force. If you have read the write up on Torque versus Horsepower you will note that a steam engine or electric motor produces maximum torque while the shaft or rotor is held stationery.  A jet engine or rocket is producing no work whilst it is stationary if we look at work =  force x distance (or displacement, ‘s’). This is what makes drawing a parallel between thrust and horsepower so compelling.

Yet, I stand corrected here, Apollo 11 developed an equivalent of something like 95 to160 million horsepower to get out of the earth’s atmosphere – I am going back in time here, Guiness Book or World Records and/or NASA. F1 engine picture found here. (see bottom before comments). Just looking at the engine tells me this is something you want strapped to your car.

Thrust: Newton’s 3rd Law

The operation of a rocket is based on conservation of linear momentum or more simply put, for every action there is an equal and opposite reaction – as the exhaust gases rush out in one direction, the rocket moves forward in the opposite direction. Once it is moving it is doing work. A jet engine may be running at maximum thrust while the pilots have the aircraft braked and because the aircraft is not moving forward no work is done. A propeller powered aircraft however may have an internal combustion engine which has a shaft coupled to the propeller which means there will be torque and as  torque is defined as the force applied at a radius from a given point, work is done.

Power available, Pushing and Pulling forces and Velocity

Because we know that Power = Force x Velocity the formula Pa = Ta V / 325 can be used, where Pa = Propulsive Power (HP), Ta = Thrust in Pounds and V = Velocity in Knots (one knot = 1852 meters or 2025.372 yards per hour). A 747 may have 60 000 pounds of thrust at 500 knots which will equal to (60^3 x 5^2)/325 = 92 307HP per engine. The most important parameter here is that the aircraft is moving at about 580 mph (not too safe). At standstill it is developing (60^2 x 0)/325 = 0 HP. No work!

I read on a forum somewhere where a comparison is then made between a jet turbine (stationary) is powering a power station. Here we have the following: gearbox and alternator. There is a force acting on the shaft (torque). Depending on the load current and at a fixed rotational speed in RPM we have a constant voltage. P in kW = voltage x current. Divide this by 0.746 and we have the power generated.

Thrust – Kort nozzles, Bollard pull and Voith Schneider

All in all, conversion from thrust to horsepower or horsepower to thrust is not a viable metric because of the dynamics we need to take into account. Once the jet engine, in our case, is actually doing work, meaning there is a force and a velocity things may assume to become simpler but my rationale tells me that we then have to redesign the way we determine what ‘doing work’ means, just like why ocean going tugs use bollard pull and not energy in Joules.  Pushing against a wall which won’t budge for ten hours is not doing work although you may feel a wreck afterwards. Tug boats use unique and innovative means to direct the thrust, just like VTOL aircraft.

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