91Ó°ÊÓ

Kinetic energy is the energy an object possesses due to its motion. For any moving object, this energy can be calculated using the formula:

• \( KE = \frac{1}{2} mv^2 \)

Where \( m \) is the mass of the object and \( v \) is its velocity. The faster an object moves, or the heavier it is, the more kinetic energy it has. In the context of the ballistic pendulum, when the bullet is fired, it starts with an initial kinetic energy based purely on its mass and velocity. As worked out in the solution, a bullet with a mass of 12 grams traveling at a speed of 380 meters per second has a kinetic energy of 867.6 Joules. This high kinetic energy is why bullets can penetrate objects and immediately transfer energy upon impact. Kinetic energy's dependence on both mass and speed implies that doubling the speed of an object can quadruple its kinetic energy, highlighting its sensitivity to velocity changes.

The principle of conservation of momentum is crucial in understanding collisions like the one between the bullet and the pendulum. It states that the total momentum of a closed system remains constant if it is not subjected to external forces. When the bullet hits the pendulum, they stick together, and their total momentum before and after the collision stays the same. This principle can be mathematically expressed as:

• \( m_{bullet}v_{bullet} = (m_{system})v_{system} \)

In our exercise, the bullet initially carries all the momentum, calculated by multiplying its mass and velocity. After the collision, the bullet and pendulum move as a combined system. By equating their pre- and post-collision momenta, we solve for the velocity of the combined system, which turns out to be 0.7598 m/s. This example exemplifies how momentum conservation aids in determining the changes in motion post-collision, a technique often used in physics to solve intricate problems.

Potential energy is the stored energy of an object due to its position or arrangement. When dealing with vertical movements under the influence of gravity, the potential energy at a given height can be defined as:

• \( PE = mgh \)

Where \( m \) is the mass, \( g \) is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and \( h \) is the height above the reference point. Following the collision, as the bullet-pendulum system rises to its highest point, kinetic energy is converted entirely into potential energy. Therefore, the initial kinetic energy of the combined system becomes equal to its potential energy at the peak of the swing. From our solution, it was shown that the kinetic energy of 1.733 Joules is converted to potential energy, enabling us to deduce that the pendulum rises to a height of approximately 0.0294 meters. Potential energy is fundamental in predicting how much an object can rise or fall within a gravitational field when kinetic energy transforms.