The gauge pressure is the net pressure acting on the tyre. I think the pressure that must be used in the equation is actually the gross pressure in the tyre, e.g., p_1+p_a.
Does this give a better answer?
The horizontal velocity of the combined mass immediately before it interacts with the inclined plane is the final velocity you found by momentum conservation. But it's likely you must find the horizontal velocity due to the plane.
To do this you must decompose the final velocity into its...
Your attachment is still pending approval as I write this.
Another approach is to multiply both sides by (1+t^2)^1/2 and observe that the LHS is the derivative, with respect to t, of a product. The integration becomes straight forward.
To anyone familiar with the Ranque-Hilsch vortex tube:
Say if the cooled air is allowed to pass over the vortex section of this tube, what would happen?
Would the effect of cooling be the destruction of the vortex? or would it act to further decrease the temperature of the cold exhaust?