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).

1. 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

2. 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)