Problem Statement:

A Boeing 747-400 has a maximum takeoff mass of 396,890 kg (910,000 lb). It is powered by four Pratt & Whitney PW4062 turbofans that produce a maximum thrust of 275 kN each. What is the initial acceleration of the 747 as it starts its takeoff role? If it maintains this acceleration, how long will it take to reach the takeoff speed of 290 km/h (180 mph)? Note: The takeoff speed is the speed required for the wings to lift the aircraft into the air.

Solution:

We can find the acceleration from Newton's 2nd Law:

But first, we need to know the total thrust produced by the engines. Using the given information we have

Let's assume friction resisting the turning of the aircraft wheels and the resistance from aerodynamic drag are small so that the net force accelerating the aircraft is equal to F_T. Then,

If the aircraft maintains a constant acceleration, we can find how long it takes to reach the takeoff speed using the definition of acceleration. Assuming the aircraft starts from the rest we have

Solving for t_takeoff

In reality, it will take a longer than this for the aircraft to reach takeoff speed because thrust is reduced as air enters the engine at a higher speed (i.e. V₀ in the thrust equation increases) and drag increases as the aircraft moves faster.