To exit the tube, the vehicle will pass through an already open fast-acting door first, and that door will start closing immediately. The other end of the airlock is covered with a burst disk. The ambient air pressure at the airlock’s altitude (15km) is around 12000 Pa and the pressure inside the tube is 5 Pa. When the vehicle breaks through the burst disk, the rarified outside air will start travelling into the tube at the speed of sound. The fast-acting door needs to finish closing before the ambient air rushing into the tube reaches it. The math in the model estimates that to meet these requirements the airlock needs to be at least 288 m long if the fast-acting door is engineered to close in 1 second. I should add that the fast-acting door can be backed by a second slower door that is designed to achieve a better vacuum seal.