Final answer:
It hurts more when snapped with a ruler because of the ruler's greater stiffness and ability to store and release more elastic potential energy quickly, resulting in a higher force and pressure applied to the hand over a shorter time period.
Step-by-step explanation:
When your hand is snapped with a ruler rather than with a loose spring, even if both the ruler and the spring are displaced by the same amount, the pain experienced is generally greater with the ruler. This difference in pain sensation is due to the stiffness of the ruler compared to that of the loose spring. A ruler is more rigid, and thus it can apply a larger force over a shorter period of time when it snaps back. This is because the ruler, having a higher elastic modulus, stores more elastic potential energy when bent, which is then released quickly as the ruler straightens. In contrast, a loose spring has a lower stiffness and a longer period of force application, resulting in a softer impact.
The concept of impulse, which is the product of force and the time over which it acts, also plays a role. Even if the displacements are the same, the more rigid ruler delivers its energy in a shorter time, meaning a higher force is applied, leading to more pain. In essence, the intensity of the force, and consequently the pressure (force per unit area) from the ruler's edge, contributes to the greater pain felt.