91Ó°ÊÓ

Kinetic energy is a fundamental concept in physics that describes the energy an object possesses due to its motion. It is calculated using the formula:

• \( KE = \frac{1}{2} mv^2 \)

where:

• \( m \) is the mass of the object (in kilograms), and
• \( v \) is the velocity of the object (in meters per second).

In our problem, we consider a bullet with mass 0.005 kg traveling at an initial speed of 450 m/s.
Substituting these values into the kinetic energy formula gives us:

• \( KE_i = \frac{1}{2} \times 0.005 \times (450)^2 = 506.25 \) Joules.

This calculation shows us the enormous energy contained in the bullet due to its high speed.
Understanding kinetic energy is crucial when analyzing how energy is transformed during motion, especially in collision scenarios like this one.

Potential energy represents the stored energy of an object due to its position within a force field, commonly a gravitational field. The formula for gravitational potential energy is:

• \( PE = mgh \)

where:

• \( m \) is the mass of the object (in kilograms),
• \( g \) is the acceleration due to gravity (approximately 9.81 m/s² on Earth), and
• \( h \) is the height the object is raised (in meters).

In the given scenario, when the bullet strikes the block, it causes the block to rise, increasing its potential energy.
Using the values from the problem, the mass of the block is 1.00 kg, the rise in height \( h \) is 0.0045 m (converted from 0.450 cm), and substituting into the equation gives us:

• \( PE = 1.00 \times 9.81 \times 0.0045 = 0.044145 \) Joules.

This potential energy determines how high the block rises due to the kinetic energy transferred from the bullet.

The law of conservation of energy is a key principle in physics which states that energy cannot be created or destroyed; it can only be transformed from one form to another.
In collisions, analyzing how energy is transferred between objects can reveal vital insights. For our bullet-block system, the conservation of energy principle helps us calculate the bullet's velocity after exiting the block.

• The initial total energy equals the sum of all forms of energy after the event.

In this exercise, the initial kinetic energy of the bullet transforms into the kinetic energy of the bullet after the collision and the potential energy of the raised block.

• Using the conservation equation: \( KE_i = KE_f + PE \), we find the kinetic energy remaining in the bullet after transferring some energy to the block.

This results in the final kinetic energy, \( KE_f \), from which we can determine the bullet's final speed. The steps included solving:

• Remaining kinetic energy: \( KE_f = 506.25 - 0.044145 = 506.205855 \) Joules.
• Using \( KE_f = \frac{1}{2} mv_f^2 \) proceeds to find the bullet’s new velocity, \( v_f \approx 449.98 \) m/s.

Thus, energy conservation provides the framework for understanding the transformation and conservation of energy in physical processes.