Final answer:
The spinothalamic tract is responsible for conducting pain and temperature sensations and would be the tract involved if a soldier doesn't feel pain during an injury in battle, allowing them to run to safety.
Step-by-step explanation:
When a soldier in battle is injured but does not feel pain, it is the spinothalamic tract that is likely affected. This tract is primarily responsible for conducting pain and temperature sensations and decussates (crosses over) in the spinal cord shortly after entering it. As pain fibers from the site of injury cross over to the opposite side in the spinal cord before ascending to the brain, an injury to the spinothalamic tract could result in an impairment or loss of pain sensation on the opposite side of the body relative to the injury.
On the other hand, the dorsal column system is primarily responsible for transmitting sensations of touch and proprioception and crosses over at a higher level in the medulla. Therefore, when considering an injury to the spinal cord tracts and the associated loss of pain sensation that allows a soldier to run to safety, it is the spinothalamic tract that plays a key role.