What numbers do we examine when predicting how much damage an object will do when it strikes another object? A 1,000 pound boulder at rest is pushed with a constant acceleration for 6 seconds, causing it to move 30 feet. It then strikes a medieval castle wall made of stone and mortar. If we know nothing more about the compositions of the wall and the boulder, what numbers should be considered when predicting how much damage the boulder will cause to the wall?
Specifically, do we care about the momentum of the boulder, the kinetic force imparted to the wall, the sectional density of the boulder, or something else? Or what combination of numbers should we examine?
 A: When one predicts the damage capacity of an object with respect to another, many factors are considered. In the typical case of an inelastic collision, a collision in which kinetic energy is not conserved, momentum and velocity are considered; the law of conservation of momentum must be followed. Kinetic energy, $1/2mv^2$, as a result, is considered due to its proportionality to momentum and velocity. 
Attached is a diagram of an inelastic collision:

A: In order to work out how much damage it can do, you look at the kinetic energy (KE$={1\over 2} m v^2$). The damage will effectively involve the breaking of chemical bonds and so part of the KE of the boulder can be used to do this.  In a completely elastic collision, no damage would be done.
A: Energy by definition is the ability of a body (or mass) to do work or make a change. In your proposed scenario, damage is the change. And so from the definition, one can say that the kinetic energy of the boulder is the first number to look at when trying to predict the damage.
That being said, looking at kinetic energy alone is not an accurate indicator of the damage because not all the energy will be transfered or "coupled" to the targeted body. And this is where many factors come into play and things get complicated.
The first factor that comes to mind is the density and hardness of the thrown object relative to the target. This is because a harder and denser object will deform less and so less energy in expended in its deformation leaving more energy available to be transfered to the target.
The second factor that is important is the cross-section area of the object. A narrower object focuses its energy transfer to a smaller volume, and so it can cause more damage.
The third factor is momentum. You can think of momentum as the difficulty to bring a moving object to a stop. So for example if you have two objects of different masses but equal kinetic energy, the object that has the higher momentum will be more difficult to stop. In ballistics, this translates to more penetration. I am not sure how this might work in case of a boulder hitting a wall, but in case of two boulders of equal kinetic energies, I would always trust the one with the higher momentum to cause more damage.
I am sure there are many more factors that should be taken into consideration to determine the amount of energy that will be imparted to the target. They can range in complexity from material properties to the simplicity of something like the angle of incidence.
So to summarize. Energy is what causes the damage, but there are many other factors that determine how much energy is actually transfered to the target, and hence energy alone can't be used to predict the damage.  
