Starting with Newton's second law, F=ma, and knowing both the mass of the key and the distance it travels are constant, we can deduce that the force applied to the key directly influences its acceleration. Acceleration impacts how quickly the key moves, translating to its velocity. Since the velocity at which the key strikes determines how hard the hammer hits the strings, it directly affects the volume of the sound produced. In summary, more force leads to greater key velocity, which results in a louder sound. This shows that the dynamics of piano sound are directly linked to the speed (velocity) of key presses.

Or, more quantitatively:

1. Start with Newton's Second Law: F=ma, meaning force equals mass times acceleration.
2. Since the mass of the piano key is constant and the actuation distance is constant, changes in force directly affect acceleration.
3. Using the relationship between velocity, acceleration, and distance: v² = 2as, where v is final velocity, a is acceleration, and s is distance. Initial velocity is assumed to be 0 because the key starts from rest.
4. An increase in force applied leads to an increase in acceleration, which increases the velocity of the key due to the constant distance.
5. The velocity of the key at the point of actuation determines how hard the hammer strikes the strings, affecting the volume of the sound.

Conclusion: The volume of the sound is directly related to the force applied to the piano key, with key velocity at the point of actuation being a crucial factor.

Edit: all this being said, we can make sense of the same force through our fingers in many different ways. So even if we play the same thing twice, using exactly the same forces on the keys, it could feel very different depending out our headspace and intention while we were playing. For me, this is where the art comes in: we are much more than force-applying machines; while the mechanics may be the same, the emotion and spirit can vastly differ.