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 back- ward force due to air resistance.)

Explanation / Answer

given the blimp is initially at rest
now force between the propellor and the air is given by
F = kt
also, force on the blimp is the same
so F = kt
now, mass of blimp = m
so from newtons second law
m*dv/dt = kt
m*dv = kt*dt
integrating from v = 0 to v = v, and t = 0 to t = t
mv = kt^2/2
but v = dx/dt
m*dx/dt = kt^2/2
2m*dx = kt^2*dt
integrating from x = 0 to x = x
t = 0 to t = t
2mx = kt^3/3
hence
x = kt^3/6m
this position over time is calculated assumiung that there is not significant aerodynamic drag

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