Date: 201605
Local time of day: 0001-0600
Location: ZZZ.Airport, US
Flight conditions: VMC
Light: Dawn
Aircraft: Large Transport; Low Wing; 2 Turbojet Eng
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Takeoff / Launch; Taxi
ATC / advisory: Tower ZZZ
Aircraft component: Main Gear Tire
Component problem: Failed
Reporter organization: Air Carrier
Reporter function: Captain; Pilot Not Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP)
Human factors: Situational Awareness
Second reporter function: Pilot Not Flying; First Officer
Anomaly: Aircraft Equipment Problem Less Severe; Ground Event / Encounter Object
Detector: Person Dispatch; Person Air Traffic Control; Person Flight Attendant; Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Provided Assistance; Aircraft Aircraft Damaged; Flight Crew Landed in Emergency Condition; Flight Crew Requested ATC Assistance / Clarification
Contributing factors: Aircraft
Primary problem: Aircraft

Narrative: On taxi out to the Runway as we turned the corner; we felt a bump; which I thought was a seam or uneven pavement. The flight attendants even called to ask what that sound was (I told them we thought it was just a bump in the pavement). As we taxied forward and onto the runway all indications were normal. It was the First Officers leg. Takeoff roll was normal until the last third; when we felt increasing vibration. Being too fast for an abort; we continued the takeoff.Liftoff was normal as was gear retraction. We discussed the vibration on climb out as a possible blown tire; but weren't sure and elected to continue to [our destination] as everything was normal at that point. Later on in flight; the flight attendants indicated that the 'bump' was quite loud in the main cabin; which had not been conveyed to us at the time of the incident.About 45 minutes into the flight we received a message from Dispatch about possible debris left on the runway. After multiple messages back and forth; we planned to [advise ATC] on arrival for a blown tire. We thought it might be a nose tire and used the QRH to plan for a landing with flat nose tires.[Advised ATC] on descent and got vectors to the ILS with a 15 mile final. I elected to let the First Officer continue to fly the approach and landing. Planned a flaps 30 landing; manual brakes and manual speed brake. We felt no need for a flyby as there was no problem when we lowered the gear early. On short final; Tower told us there were no gross visual indications of gear problems so we elected to land. Airport Rescue and Fire Fighting (ARFF) equipment followed us down the runway and; after a full stop on the runway; notified us that it was the number 3 tire that was blown. No other problems noted by the fire department; so we elected to taxi to the gate with no further incident.I should have solicited more feedback from the flight attendants on what they had experienced when we 'hit the bump'. Then I might have realized that it was more than a bump and would have sought external inspection of the aircraft before takeoff. I didn't realize what a difference in noise level between the cockpit and the cabin when a tire pops.


Second narrative: [Report narrative contained no additional information.]


Synopsis: Air carrier flight had a main gear tire fail during taxi out unknown to the crew. Later detection by Tower; flight attendants; and Dispatch concluded that a tire had failed. Crew landed normally with the assistance of airport resources.