A blimp is initially at rest, hovering, when at t = 0 the pilot turns on the eng

Question

A blimp is initially at rest, hovering, when at t = 0 the pilot turns on the engine driving the propeller. The engine cannot instantly get the propeller going, but the propeller speeds up steadily. The steadily increasing force between the air and the propeller is given by the equation F = kt, where k is a constant. If the mass of the blimp is m, find its position as a function of time. (Assume that during the period of time you’re dealing with, the blimp is not yet moving fast enough to cause a significant backward force due to air resistance.)

Explanation / Answer

Given force =F=kt

We know from Newton's law that force=ma

a=F/m=kt/m

Also, d²x/dt² =a where x is position

With initial conditions x(0)=dx/dt(0)=0,we can solve the differential equation

d²x/dt²=kt/m

dx/dt=kt²/2m

x=kt³/6m

Therefore, position of blimp as a function of time is

x=kt³/6m

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