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Revealing mistake: When Jack is hanging out of the truck you can see him standing on the hood of the car which Lydia is driving. At some points we get camera view from the back seat of the car. In some angles the mirror inside the car disappear. This is to prevent us from seeing the camera. (01:04:15)
Answer: Probably, but most likely it's because Jack's pod is fragile and wouldn't survive the shock wave heartbeats. The pod is experimental and built for scientific observation, not battle.