91Ó°ÊÓ

In physics, the conservation of momentum is a crucial principle. It states that in a closed system, the total momentum is the same before and after an event. This means that no matter what happens within the system, the momentum remains unchanged—as long as no external forces act on it.
In our scenario where a bullet is fired into a block, the system consists of the bullet and the block. The bullet has an initial momentum, calculated as its mass (\(m\)) multiplied by its velocity (\(v_0\)).
When the bullet lodges into the block, the combined system of bullet and block move with a common velocity. The sum of their masses is \((M + m)\), with the final velocity being \(v = \frac{m \cdot v_{0}}{M + m}\).

• The total momentum before collision: \(m \cdot v_0\)
• The total momentum after collision: \((M + m) \cdot v\)

By equating these two values, the principle of conservation of momentum allows us to relate the initial speed of the bullet with the speed of the block and bullet after the collision.

Friction is the resistive force that opposes the motion of objects. It can convert kinetic energy into other forms, such as heat, preventing perpetual motion. This exercise involves friction because the block, with the embedded bullet, slides to a stop.
Sliding friction is quantified by the coefficient of friction \(\mu\) and is constant when the block moves. The frictional force can be calculated as \(\mu \cdot M \cdot g\), where \(M\) is the block's mass and \(g\) is the acceleration due to gravity. It steadily works against the block's kinetic energy, bringing it to rest after sliding a distance \(d\).

• This frictional force does negative work on the block, \(-\mu \cdot M \cdot g \cdot d\).
• This work against friction enables us to connect the kinetic energy immediately post-collision with the stopping distance of the block.

This connection between friction, energy conversion, and motion distance is vital for understanding how the energy in motion is dissipated.

Kinetic energy is the energy that an object possesses due to its motion and is given by the equation:\[KE = \frac{1}{2} m v^2\]
In our scenario, once the bullet embeds in the block, the whole system (block plus bullet) has kinetic energy determined by their combined mass and velocity:\[\text{KE} = \frac{1}{2} (M + m) v^2\]
During the bullet's embedding and block’s sliding, the initial kinetic energy in the system is transformed, primarily into thermal energy due to friction. The work done by friction equals this reduction in kinetic energy:

• At the start, the kinetic energy is \(\frac{1}{2} (M + m) v^2\), representing the system’s motion post-collision.
• As the block slides to a stop, the kinetic energy becomes zero.

By balancing the kinetic energy and the frictional work, the equation \(\frac{1}{2} (M + m) v^2 = \mu \cdot M \cdot g \cdot d\) helps solve for the initial speed \(v_0\) of the bullet.

The work-energy principle states that the net work done on a system is equal to the change in its kinetic energy. This principle is an essential concept in understanding how forces lead to changes in an object’s velocity.
In our exercise, after the bullet is lodged in the block, friction does work on the moving block-bullet system. Here’s a breakdown:

• The work done by friction is negative because it acts opposite to the motion.
• Initially, the moving block-bullet system has kinetic energy because of its velocity \(v\).

As friction performs work, it removes energy from the system, eventually bringing it to a halt with zero kinetic energy:\[\text{Work} = \Delta \text{KE} = \text{-}\mu \cdot M \cdot g \cdot d = \frac{1}{2} (M + m) v^2 - 0\]
Applying the work-energy principle, you can link the distance the block slides to the original bullet speed, demonstrating how energy initially in motion is dissipated through frictional work.