Which day during an typical Indy Car race weekend do the drivers sign autographs and take photos?

I can't speak for other races, but I do know that leading up to the Indy 500 race they sign autographs on the Wednesday before the race (aka Community Day), and sometimes on Fridays/Saturdays at retail stores like Meijers, Krogers, etc.

Since Barber is a Sunday road course race, the official driver free autograph session will probably be on Saturday. Depending on how the schedule is laid out, it could be either before or after qualifying. However, for a normal Sunday race, the autograph session is always on a Saturday. FWIW, the event timing schedule should be released soon for Barber. Keep an eye on the race's page at indycar.com for about the next month or so. They always put up a PDF schedule on their individual race pages that defines when everything is supposed to happen on that race's weekend. I see no reason for Barber to be any different in that.

Check the weekend race schedule and the individual driver websites. These will provide the dates, times, and locations for autograph sessions. As for meeting the drivers, if you get a garage pass you can meet them but they may not be very interested in chatting and/or signing the closer you are to race day. If you are trying to meet a driver in the garage try approaching them on a Friday when they are not in the final prep for race day.

If you can go on qualifying day, my friends on the east coast they always go on qualifying day, they've hit the autograph jackpot a couple times with more than 10 drivers signing programs, or caps and t-shirts. You can also check each drivers websites for locations they might be at before the race.

Go Team Penske

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it just depends .. some have autograph sessions the day of and the day before usually at one of their haulers

i was lucky last year when i got a video of Franchitti and Dixon at Chicagoland this was the day of the race