when a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. this is accompanied by a contraction of the trachea, making a narrower channel for the expelled air to flow through. for a given amount of air to escape in a fixed time, it must move faster through the narrower channel than the wider one. the greater the velocity of the airstream, the greater the force on the foreign object. x rays show that the radius of the circular tracheal tube contracts to about two-thirds of its normal radius during a cough. according to a mathematical model of coughing, the velocity v of the airstream is related to the radius r of the trachea by the equation