Imagine a spherical balloon filled wild helium the balloon i

Imagine a spherical balloon filled wild helium- the balloon itself is magically crafted to expand and contract freely without exerting any restorative force Some weights (with negligible volume) are hung from the balloon such that the total mass of helium + balloon + weights is m Assume the balloon is surrounded by with constant density p_0 and constant pressure P_0, independent of altitude.

Solution

Gas equation   PV = nRT

As the pressure P is constant

P dV/dt = nR dT/dt

dV/dt =nR/P dT/dt

initial condition    P₀V₀ = nRT₀

dV/dt = (V₀/T₀) dT/dt

upward thrust on the ballon

F = Vg – mg

Differentiate with t , we have

dF/dt = gdV/dt

initial the balloon is at rest hence

V₀ g – mg = 0 hence m = V₀

mda/dt = gdV/dt

                = g*(V₀/T₀)* dT/dt

da/dt = g/T₀ * dT/dt

we can express rate of change of acceleration in terms of g, which has units of acceleration

da/dt = (1/T₀)dT/dt

integrating it

a(t) = T(t)/T₀

given T(t) = T₀(2 - 1/t+1)

a(t) = 2 – 1/t+1

dv(t)/dt = 2 – 1/t+1

v(t) = 2t – ln(t+1)