91Ó°ÊÓ

A \(0.500-\mathrm{kg}\) sphere moving with a velocity \((2.00 \hat{\mathbf{i}}-3.00 \hat{\mathbf{j}}+$$1.00 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s}\) strikes another sphere of mass \(1.50 \mathrm{kg}\) moving with a velocity \((-1.00 \hat{\mathbf{i}}+2.00 \hat{\mathbf{j}}-3.00 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s}\) (a) If the velocity of the \(0.500-\mathrm{kg}\) sphere after the collision is \((-1.00 \hat{\mathbf{i}}+3.00 \hat{\mathbf{j}}-8.00 \mathbf{k}) \mathrm{m} / \mathrm{s},\) find the final velocity of the 1.50-kg sphere and identify the kind of collision (elastic, inelastic, or perfectly inelastic). (b) If the velocity of the \(0.500-\mathrm{kg}\) sphere after the collision is \((-0.250 \hat{\mathbf{i}}+0.750 \hat{\mathbf{j}}-\) \(2.00 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s},\) find the final velocity of the \(1.50-\mathrm{kg}\) sphere and identify the kind of collision. (c) What If? If the velocity of the \(0.500-\mathrm{kg}\) sphere after the collision is \((-1.00 \hat{\mathbf{i}}+3.00 \hat{\mathbf{j}}+\) \(a \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s},\) find the value of \(a\) and the velocity of the \(1.50-\mathrm{kg}\) sphere after an elastic collision.

Step by step solution

01

Analyze the Initial State and Apply Conservation of Momentum

The total initial momentum is given by: \[ \vec{P_{i}} = m_1 \cdot \vec{v_{1i}} + m_2 \cdot \vec{v_{2i}} \] while the total final momentum is given by: \[ \vec{P_{f}} = m_1 \cdot \vec{v_{1f}} + m_2 \cdot \vec{v_{2f}} \] Apply the conservation of momentum, which states that the total momentum before the collision equals to the total momentum after the collision. This gives the equation: \[ \vec{P_{i}} = \vec{P_{f}} \] where \(m_1\) and \(m_2\) are the masses, \(\vec{v_{1i}}, \vec{v_{1f}}, \vec{v_{2i}}, \vec{v_{2f}}\) are the initial and final velocities.

02

Calculate the Final Velocity of the Second Sphere

Isolate \(\vec{v_{2f}}\) from the conservation of momentum equation. Substitute \(m_1\), \(m_2\), \(\vec{v_{1i}}\), and \(\vec{v_{1f}}\) to find \(\vec{v_{2f}}\). Repeat this process for each sub-question.

03

Identify the Type of Collision

Apply the properties of each type of collision to the results. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, only momentum is conserved, kinetic energy isn't. In a perfectly inelastic collision, momentum is conserved and the objects stick together after the collision. Calculate the total initial and final kinetic energy and compare:

04

Solve for the Error Case

In case a) if both the momentum and kinetic energy are conserved, it's an elastic collision. If only momentum is conserved, it's an inelastic collision. In case b) follow the same method. In case c), since we're told it's an elastic collision, we can solve for a in the conservation of kinetic energy equation after calculating the value of \(\vec{v_{2f}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Collision Types

When two objects collide, their interaction can be classified into different types based on the conservation laws applicable. Understanding these helps us predict the outcome of a collision.

• Elastic Collision: Both momentum and kinetic energy are conserved. Neither object is deformed, and no energy is lost to sound or heat. This ideal condition rarely occurs in real-world scenarios.
• Inelastic Collision: While momentum is conserved, kinetic energy is not. Some energy is converted into sound, heat, or deformation. It's a more common occurrence in everyday life.
• Perfectly Inelastic Collision: A special type of inelastic collision where the colliding objects stick together after impact, moving as a single mass.

These outcomes depend on initial velocity, mass, and the nature of the materials involved in the collision.

Elastic Collision

An elastic collision is a perfect scenario in physics where no kinetic energy is lost. Both momentum and kinetic energy are conserved during the impact. For an elastic collision:

• The sum of the initial momenta of the colliding bodies is equal to the sum of their final momenta.
• The total kinetic energy of the system before and after the collision remains constant.

In practice, perfectly elastic collisions are rare, as they require ideal conditions, where none of the kinetic energy converts into other forms such as heat or sound.
However, they can be closely approximated in environments like billiards or physics simulations, where friction and other external forces are minimal.

Inelastic Collision

In contrast to elastic collisions, inelastic collisions do not conserve kinetic energy. In these cases, some portion of the initial kinetic energy is transformed into other energy forms, such as sound or heat, or goes into deforming the bodies involved.

• Momentum is still conserved, meaning the total momentum before and after the collision remains the same.
• Kinetic energy loss is often perceived through the heating of objects or sound emitted upon impact.

A common example of inelastic collisions is a car crash, where significant energy is spent in deforming the cars, creating sound, and other forms of energy dissipation.
Understanding these differences is crucial in many applications, from engineering safety features in vehicles to predicting the outcomes of atomic collisions in physics.