Question

A billiard ball moving with a speed of \(8 \mathrm{~m} / \mathrm{s}\) collides with an identical ball originally at rest. If the first ball stops after collision, then the second ball will move forward with a speed of .... (elastic collision) (A) \(8 \mathrm{~m} / \mathrm{s}\) (B) \(4 \mathrm{~m} / \mathrm{s}\) (C) \(16 \mathrm{~m} / \mathrm{s}\) (D) \(1.0 \mathrm{~m} / \mathrm{s}\)

Short answer

The second ball will move forward with a speed of \(8 \mathrm{~m} / \mathrm{s}\) (A).

Step-by-step solution

Conservation of linear momentum

In an elastic collision, the linear momentum is conserved. We can write the conservation of linear momentum as: \(m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}\), where \(m_1\) is the mass of the first ball, \(m_2\) is the mass of the second ball, \(v_{1i}\) is the initial velocity of the first ball, \(v_{2i}\) is the initial velocity of the second ball, \(v_{1f}\) is the final velocity of the first ball, and \(v_{2f}\) is the final velocity of the second ball. Since the two balls are identical, their masses are equal, so \(m_1 = m_2 = m\). Also, we know the initial velocities of both balls: \(v_{1i} = 8 \mathrm{~m} / \mathrm{s}\) and \(v_{2i} = 0\). The final velocity of the first ball is also given: \(v_{1f} = 0\). Now our equation becomes: \(mv_{1i} + 0 = 0 + mv_{2f}\)

Solving for the final velocity of the second ball

Now we can solve the equation for the final velocity of the second ball, \(v_{2f}\): \(mv_{1i} = mv_{2f}\) Since both masses are identical, we can divide both sides of the equation by m: \(v_{1i} = v_{2f}\) Substituting the given value of \(v_{1i}\): \(8 \mathrm{~m} / \mathrm{s} = v_{2f}\) So, the final velocity of the second ball is 8 m/s. Therefore, the correct answer is: (A) \(8 \mathrm{~m} / \mathrm{s}\)

Key concepts

Linear Momentum Conservation

In physics, the principle of linear momentum conservation is a fundamental concept that applies to various scenarios, especially in collisions.
It states that in a closed system with no external forces acting on it, the total linear momentum before an event is equal to the total linear momentum after the event.
Mathematically, this principle is expressed as:

• \(m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}\)

In this equation:- \(m_1\) and \(m_2\) are the masses of the two objects involved.- \(v_{1i}\) and \(v_{2i}\) are their initial velocities.- \(v_{1f}\) and \(v_{2f}\) are their final velocities.
For our exercise, when a moving billiard ball collides with another identical ball at rest, the total momentum at the start equals the total momentum after the collision, ensuring that we can solve for the unknown velocities.

Billiard Ball Collision

Billiard ball collisions offer a fascinating insight into elastic collisions. In games like pool, observing a cue ball collide with other balls provides a practical demonstration of complex physics principles.
When a moving billiard ball impacts another ball at rest, the type of collision can be defined as elastic if the energy is conserved along with momentum. In the context of billiard balls:

• Elastic collisions ensure no kinetic energy is lost to sound, deformation, or heat.
• The velocities are exchanged depending on certain conditions, like mass equivalence and initial rest states, as explored in this scenario.
• Due to their hard surface, billiard balls exhibit nearly perfect elastic collisions, making them excellent teaching examples.

Understanding these dynamics not only helps in mastering physics problems but also in improving strategic gameplay in billiards itself.

Velocity Exchange

Velocity exchange is a unique occurrence in some collisions where two objects swap their velocities. This phenomenon prominently occurs in elastic collisions involving identical objects.
Imagine two identical billiard balls, one moving and one at rest. On collision, the moving ball halts, and the initially stationary ball moves with the velocity of the first ball.

• This is due to the conservation laws ensuring total momentum and energy preservation.
• The exact exchange occurs because the masses are identical, and no external forces intervene.

In our problem, the first ball stops completely post-collision, granting its entire velocity to the second ball. Thus, the exchange of velocity results in the second ball moving with the same speed of 8 m/s as initially held by the first ball before the impact.

Identical Masses Collision

A collision involving identical masses is a simplified yet intriguing instance of elastic collisions. When two objects with the same mass meet, the problem's complexity often diminishes due to symmetry.
In such scenarios, analyzing momentum and energy conservation becomes more straightforward.

Key Outcomes of Identical Masses Collision:

- Momentum conservation leads to easier calculations as mass values do not differ.- Velocity exchange occurs naturally if one object is initially at rest, simplifying the dynamics significantly.- Predictions of post-collision velocities become accurately attainable using simple algebraic equations.
In the example discussed, the identical mass of the billiard balls facilitated a direct velocity exchange without complex calculations, as both masses (\(m_1 = m_2\)) canceled out in the conservation equations.
These characteristics highlight the elegance and predictability of collisions with identical masses.