By using a long time to stop the ship, the force doesn't have to be as large, which is good because the engines can only supply a certain amount of force. The padding in the dashboard compresses when something hits it. This takes time, thereby reducing the force needed for a given impulse. The necessary impulse is set by the change in momentum needed to bring the person to rest, which is fixed by the car s speed and the persons mass. Thick floor mats take time to compress so the force needed to change the gymnasts momentum to zero is less.
Less force means less chance of injury.
An impulse is needed to change the falling climbers momentum to zero. A stretching rope takes longer to stop the climber, so the force on the climber can be smaller for the same momentum change. In this problem the force is constant, set by the limit that the person must survive. The time of impact is different for the different surfaces. Impact on concrete has the shortest time of impact, so for a fixed force has the smallest survivable impulse and therefore the smallest change in momentum.