Question

A bullet of mass Mb is fired horizontally with speed Vi at a wooden block of mass Mw resting on a frictionless table. The bullet hits the block and becomes completely embedded within it. After the bullet has come to rest within the block, the block, with the bullet in it, is traveling at speed Vf .

1)Which of the following best describes this collision?
a)perfectly elastic
b)partially inelastic
c)perfectly inelastic

2) Which of the following quantities, if any, are conserved during this collision?
a) kinetic energy only
b) momentum only
c) kinetic energy and momentum
d) neither momentum nor kinetic energy

3)What is the speed of the block/bullet system after the collision, Vf?
Express your answer in terms of Vi, Mw, and Mb.

Answer 1

Concepts and reason

The concept required to solve this problem is elastic collision, inelastic collision, and law of conservation of linear momentum.

Initially, analyze whether the collision is elastic or perfectly inelastic. Later, find whether the momentum is conserved or not. Finally, apply the law of conservation of momentum to find the final velocity of the bullet block system.

Fundamentals

The expression for the conservation of linear momentum is as follows:

${m_{\rm{b}}}{v_{\rm{i}}} + {m_{\rm{w}}}{v_{\rm{w}}} = \left( {{m_{\rm{b}}} + {m_{\rm{w}}}} \right){v_{\rm{f}}}$

Here, ${m_{\rm{b}}}$ is the mass of the bullet, ${v_{\rm{i}}}$ is the initial velocity, ${m_{\rm{w}}}$ is the mass of water, and ${v_{\rm{f}}}$ is the final velocity.

In the elastic collision, the momentum and the total kinetic energy, and the total energy before and after collision is same.

In the inelastic collision, the momentum, total energy is conserved, but, the kinetic energy is not conserved.

The expression for the kinetic energy is as follows:

$K = \frac{1}{2}m{v^2}$

Here, m is the mass and v is the velocity.

The linear momentum is defined as the product of the mass and the velocity of an object.

The expression for the linear momentum is as follows:

$p = mv$

Here, p is the linear momentum, m is the mass, and v is the velocity.

(1)

A collision in which two objects stick together or employed into one another is called perfectly inelastic collision. In this collision, the kinetic energy of the system is not conserved and linear momentum is conserved.

A collision in which two objects do not stick together or embedded into one another is called perfectly elastic collision. In this collision, the kinetic energy of the system is conserved and linear momentum is conserved. In this collision, the objects bounce away from each other after the collision.

A collision in which two objects do not stick together or employed into one another is called partially elastic collision. In this collision, the kinetic energy of the system is not conserved and linear momentum is conserved. In this collision, the objects bounce away from each other after the collision.

(2)

From the definition of the perfectly inelastic collision, the linear momentum of the system of bullet and wooden block only conserves.

(3)

The wooden block is initially at rest. Therefore, the initial velocity of the block is equal to zero.

According to the law of conservation of linear momentum,

${m_{\rm{b}}}{v_{\rm{i}}} + {m_{\rm{w}}}{v_{\rm{w}}} = \left( {{m_{\rm{b}}} + {m_{\rm{w}}}} \right){v_{\rm{f}}}$

Substitute 0 m/s for ${v_{\rm{w}}}$ in the equation ${m_{\rm{b}}}{v_{\rm{i}}} + {m_{\rm{w}}}{v_{\rm{w}}} = \left( {{m_{\rm{b}}} + {m_{\rm{w}}}} \right){v_{\rm{f}}}$ and solve for ${v_{\rm{f}}}$.

$\begin{array}{c}\\{m_{\rm{b}}}{v_{\rm{i}}} + {m_{\rm{w}}}\left( {0.00{\rm{ m/s}}} \right) = \left( {{m_{\rm{b}}} + {m_{\rm{w}}}} \right){v_{\rm{f}}}\\\\{m_{\rm{b}}}{v_{\rm{i}}} = \left( {{m_{\rm{b}}} + {m_{\rm{w}}}} \right){v_{\rm{f}}}\\\\{v_{\rm{f}}} = \frac{{{m_{\rm{b}}}{v_{\rm{i}}}}}{{{m_{\rm{b}}} + {m_{\rm{w}}}}}\\\end{array}$

Ans: Part 1

The collision between the bullet and the wooden block is perfectly inelastic.

Part 2

The momentum only conserves during the collision.

Part 3

The speed of the block-bullet system is $\frac{{{m_{\rm{b}}}{v_{\rm{i}}}}}{{{m_{\rm{b}}} + {m_{\rm{w}}}}}$.