Final answer:
When the speed of an airplane is doubled while keeping the radius unchanged, the centripetal force acting on it becomes four times as great due to the square relationship between velocity and force in the equation F = mv²/r.
Step-by-step explanation:
If the speed of the airplane is doubled and the radius of the path remains unchanged, the magnitude of the centripetal force acting on the airplane will increases. The centripetal force is given by the formula F = mv²/r, where F is the centripetal force, m is the mass of the object, v is the velocity, and r is the radius of the circular path.
When you double the speed, the velocity term in the formula is squared, which means that the new force will be v² times m divided by r. Doubling v makes (2v)², or 4v². Hence, the centripetal force will be four times as much if the speed doubles with the radius staying constant.