Barret was riding in the back of the truck, no?
In order to shoot, he has to use his left hand to pull the trigger on his gun-arm. You can see him do this during battle, and during some in-game scenes. If he was doing that, then he'd be unable to hold onto the truck. He'd fall off the next time Tifa went careening around a corner, from then he'd have to run to keep up. That'd be awfully hard on his feet, since Barret is a big man. The repetitive impacting of sprinting down a highway like that would tire him out, and also jar his spine. The cramps and aches could last a long time, and might even form the basis for a repetitive-strain injury (RSI). Not to mention the abrasions and contusions he'd sustain while falling from the truck itself. He has a tattoo on his left shoulder; if he fell on that side, he could tear up the flesh, which would ruin the tattoo. Other injuries could be healed with curative magic, but not damage to body art like that. Barret's large but bare arms and shoulders would make him especially vulnerable to superficial but painful and bloody injuries in that area.