Question

Consider a ballistic pendulum (see Section 7.6 ) in which a bullet strikes a block of wood. The wooden block is hanging from the ceiling and swings up to a maximum height after the bullet strikes it. Typically, the bullet becomes embedded in the block. Given the same bullet, the same initial bullet speed, and the same block, would the maximum height of the block change if the bullet did not get stopped by the block but passed through to the other side? Would the height change if the bullet and its speed were the same but the block was steel and the bullet bounced off it, directly backward?

Short answer

Answer: Scenario 2, where the block is made of steel and the bullet bounces off, results in the same maximum height as in the initial situation. This is because both momentum and kinetic energy are conserved in this elastic collision, and the maximum height will not be affected if the bullet's speed and mass remain the same.

Step-by-step solution

Understand the initial scenario

Initially, the bullet is moving horizontally with a certain initial speed while the block is hanging from the ceiling at rest.

Determine the maximum height

We need to find the maximum height the combined bullet-block system reaches after the collision. To do this, we will use the conservation of momentum and energy in each scenario.

Scenario 1 - Bullet passes through the block

In this case, the bullet exchanges some of its momentum with the block but does not become embedded in it. We analyze this situation using the conservation of momentum: m_bullet * v_bullet_initial = (m_bullet + m_block) * v_system_final Since the total mechanical energy is not conserved in this inelastic collision, we cannot conclude if the maximum height would be the same as in the initial scenario.

Scenario 2 - Steel block and bullet bounce off

Now, when the block is made of steel, the bullet bounces off instead of getting embedded. This situation can be considered an elastic collision, so both momentum and kinetic energy are conserved: m_bullet * v_bullet_initial = m_bullet * (-v_bullet_final) + m_block * v_block_final Given that both momentum and kinetic energy are the same as in the initial scenario, we can conclude that the maximum height will not be affected if the bullet's speed and mass remain the same.

Key concepts

Conservation of Momentum

The principle of conservation of momentum tells us that when no external forces act on a system, the total momentum of that system remains constant. Momentum is a vector quantity, defined as the product of an object's mass and its velocity (\( p = mv \)). In a physics problem involving a collision, such as the one with the ballistic pendulum, understanding this principle is crucial.

When a bullet strikes a block, as described in the exercise, the bullet and block together constitute an isolated system, assuming no external forces are acting on it (like air resistance or additional external forces). Therefore, the momentum before and after the collision must be the same.

Understanding conservation of momentum allows us to set up equations to solve for unknowns such as the final velocity of the system after the collision (\( m_{bullet} \times v_{bullet_{initial}} = (m_{bullet} + m_{block}) \times v_{system_{final}} \text{). This concept is vital for predicting the outcomes of collision events in physics.

Inelastic Collision

An inelastic collision is characterized by the merging or joint movement of colliding objects after the impact. In these scenarios, kinetic energy is not conserved, although momentum is. This aspect is crucial to understand the situation where the bullet becomes embedded in the block of wood.

During this inelastic collision, the kinetic energy transforms into other forms of energy, such as internal energy or sound. This means that while we can use the conservation of momentum to find the final velocity of the system, we cannot directly use the conservation of kinetic energy to predict the height the block will reach thereafter.

The lack of kinetic energy conservation makes calculating the maximum height more complex, requiring further analysis of energy transformation during and after the collision.

Elastic Collision

In contrast to inelastic collisions, an elastic collision is an event where both momentum and kinetic energy are conserved. This conservation makes elastic collisions relatively simpler to analyze mathematically.

In the scenario where the bullet bounces off a steel block, we're dealing with an elastic collision. Both objects remain separate after the collision, and the kinetic energy remains within the system. Here, we can set up equations that reflect the conservation of kinetic energy in addition to momentum:

• Momentum conservation equation: \( m_{bullet} \times v_{bullet_{initial}} = m_{bullet} \times (-v_{bullet_{final}}) + m_{block} \times v_{block_{final}} \)
• Kinetic energy conservation equation: \( \frac{1}{2}m_{bullet}v_{bullet_{initial}}^2 = \frac{1}{2}m_{bullet}v_{bullet_{final}}^2 + \frac{1}{2}m_{block}v_{block_{final}}^2 \)

By solving these equations, we can affirm that the block's max height will not change if the bullet's mass and speed do not change, because the system's total mechanical energy is conserved before and after the collision.

Kinetic Energy Conservation

Kinetic energy conservation is a fundamental concept in physics that applies to certain types of collisions — specifically elastic collisions. Kinetic energy, expressed as \( \frac{1}{2}mv^2 \), represents the energy an object has due to its motion.

In our exercise, we're asked to consider whether the maximum height of a swinging block changes under different collision conditions. If kinetic energy is conserved (as in the case of an elastic collision), then the total mechanical energy of the system (potential energy and kinetic energy) before and after the collision should remain the same.

This conservation principle enables us to assert that the maximum height reached by the pendulum, which depends on the potential energy at that height, will not change if the initial kinetic energy remains the same. However, in inelastic collisions, where kinetic energy is not conserved, this direct conclusion cannot be made.