Trolley travelling along a frictionless track

Question

A trolley travels with initial speed v₀ along a frictionless, horizontal, linear track. It slows down by ejecting gas in the direction of motion. The gas is emitted at a constant mass ejection rate α and with constant speed u relative to the trolley. The trolley and its supply of gas initially have a combined mass of m₀. How much time is spent ejecting gas before the trolley stops? Assume that the trolley carries sufficient gas.

Answer 1

My answer got deleted, so I am going to summarize:

-au (the neg. momentum change of the gases = m(dv_Trolley/dt (the momentum change of the trolley)

This is separable after we sub in m = m₀ - at: dv/u = -a/(m₀ - at) dt

Integrating: (v-v₀)/u = ln|(m₀ - at)/m₀| (from v₀ to v, and 0 to t)

solving for t for v = 0:

-v₀/u = ln((m₀ -at)/m₀) I removed the absolute value as m₀ > at since it is specified that there is enough gas mass to stop the trolley

Just rearrange and solve for t.