A castle's defenders throw rocks down on their attackers from a 15-m-high wall, with initial speed \(10 \mathrm{m} / \mathrm{s}\). How much faster are the rocks moving when they hit the ground than if they were simply dropped?

Kinetic energy is the energy possessed by an object due to its motion. It's a concept that often puzzles students, but it becomes clearer once you think of it as the energy that builds up when an object speeds up. In mathematical terms, kinetic energy (KE) is given by the equation \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity. This formula shows that the kinetic energy increases with the square of the velocity; hence, even a small increase in speed results in a considerable increase in kinetic energy.

In the context of the exercise with a castle's defenders throwing rocks, when the rocks pick up speed while falling, their kinetic energy increases until the moment of impact. Initially thrown rocks have more kinetic energy when reaching the ground compared to dropped rocks, because the thrown rocks benefit from both the initial motion and the acceleration due to gravity.

The acceleration due to gravity is a constant force that pulls objects towards the center of the Earth. On the surface of the Earth, this acceleration is roughly \( 9.81 \text{m/s}^2 \). This force is what causes objects to fall to the ground when dropped. Importantly, every object, regardless of mass, experiences the same acceleration due to gravity when in freefall. This often surprises students, but it's a fundamental aspect of Newton's law of universal gravitation.

In our exercise with the rocks being thrown or dropped from a height, gravity is the force that changes the rocks' kinetic energy over the distance they fall. Even for rocks thrown upwards or sideways, gravity still acts to pull them downward, which is why we consider it in the final speed calculations. The interplay between the initial kinetic energy that the defenders provide and the relentless pull of gravity determines the rocks' final velocity.

Initial speed, or initial velocity, refers to the speed at which an object starts moving. It’s an essential value when calculating the future position or speed of an object in motion problems. A lot can be inferred about an object’s journey when you know where it started. If you’re throwing an object, like the rocks in our problem, the initial speed adds to the object's energy, which changes how the situation unfolds.

In instances where objects are dropped with zero initial speed, they only gain kinetic energy from gravity as they fall. But when they are thrown with an initial speed, they start with an initial kinetic energy that must be considered in the kinetic energy equation. So the higher the initial speed, the greater the impact when it lands. Knowing the initial speed (in the case of our castle, \(10 \text{m/s}\)) helps to predict the final speed after factoring in the gravitational pull over the height from which the rock is thrown.