Problem 51 - College Physics

Question

A billiard ball mowing at \(5.00 \mathrm{~m} / \mathrm{s}\) strikes a stationary ball of the same mass. After the collision, the first ball moves at \(4.39 \mathrm{~m} / \mathrm{s}\) at an angle \(0 \mathrm{f} 30^{\circ}\) with respect to the original line of motion. (a) Find the velocity (magnitude and direction) of the second ball after collision. (b) Was the collision inelastic or elastic?

Step-by-Step Solution

Answer

The velocity of second ball after collision and nature of the collision (elastic or inelastic) will depend on the values of the computed momentum and kinetic energy changes. Use the equations in Step 2 and Step 3 to calculate these.

Step 1: Calculate initial and final momentum

The initial momentum (before collision) of the system is given by \( p_{\text{initial}} = mv \), where \( m \) is the mass and \( v \) is the velocity of the first ball. Since the second ball is stationary, its momentum is 0. After the collision, the momentum of the first ball changes direction. Let's denote its horizontal (x-axis) component as \( p_{x1} = m*v_{1}*cos(30) \) and its vertical component (y-axis) as \( p_{y1} = m*v_{1}*sin(30) \). The final momentum of the second ball should account for the remainder to conserve momentum, so \( p_{x2} = p_{\text{initial}} - p_{x1} \) and \( p_{y2} = -p_{y1} \). These will be used to calculate the final velocity of the second ball.

Step 2: Calculate the velocity of the second ball

The velocity (magnitude and direction) of the second ball can be computed as follows: Its speed (magnitude) is given by \( v_{2} = \sqrt{(p_{x2}/m)^2 + (p_{y2}/m)^2} \). The direction, given as an angle relative to the x-axis, can be found using the tangent function. That is, \( \theta_{2} = arctan ((p_{y2}/m) / (p_{x2}/m)) \). Note that you may have to adjust the angle depending on the quadrant.

Step 3: Analyze the nature of collision

A collision can be classified as elastic or inelastic. In an elastic collision, the total kinetic energy before collision equals the total kinetic energy after collision. On the other hand, in an inelastic collision, the total kinetic energy is not conserved. Calculate the initial kinetic energy, \( KE_{\text{initial}} = 0.5*m*v^2 \) (because ball 2 is stationary), and the final kinetic energy, \( KE_{\text{final}} = 0.5*m*v_{1}^2 + 0.5*m*v_{2}^2 \). If \( KE_{\text{initial}} = KE_{\text{final}} \), the collision is elastic, otherwise it is inelastic.

Key Concepts

Momentum conservation

In physics, momentum conservation is a foundational principle that helps us understand and predict the behavior of objects during interactions, like collisions.
Momentum is a vector quantity, meaning it has both magnitude and direction. It is defined as the product of an object's mass and velocity, given by the equation:

• Momentum, \( p = m \times v \)

For a collision involving two billiard balls, like in the exercise, the total momentum before and after the collision remains the same, provided no external forces act on them.
Before the collision, one billiard ball is moving, and the other is stationary, giving zero momentum to the second ball. After the collision, momentum is distributed between the balls, conserving the initial system momentum.
The changes in the direction of the ball's momentum are accounted for by the components of momentum. We break them into x (horizontal) and y (vertical) components, especially when dealing with angles. This allows us to calculate the velocity of each ball following the collision, maintaining momentum conservation.

Elastic and inelastic collisions

When objects collide, the type of collision they undergo is categorized into elastic or inelastic collisions.

• In an elastic collision, both momentum and kinetic energy are conserved.
• In an inelastic collision, only momentum is conserved, while kinetic energy is not.

To determine the nature of the collision, we consider the kinetic energy before and after the event.
During an elastic collision, objects bounce off each other without any deformation or generation of heat, maintaining their velocity magnitude post-collision.
In inelastic collisions, some kinetic energy is transformed into other forms of energy, such as heat or sound, resulting in a loss of total kinetic energy in the system.
Establishing whether a collision is elastic or inelastic can help predict future interactions, providing insight into real-world applications like car crashes or games with bouncing balls.

Kinetic energy

Kinetic energy is the energy that an object possesses due to its motion. It is one of the key metrics used when analyzing collisions in physics. This energy can be calculated using the equation:

• Kinetic Energy, \( KE = \frac{1}{2} m v^2 \)

In the context of a billiard ball collision exercise, understanding kinetic energy helps us see how energy is transferred between the balls.
Calculating kinetic energy before and after the collision is essential to decide if a collision is elastic or inelastic.
If the kinetic energy values before and after are equal, the energy is preserved, classifying the collision as elastic. If they differ, it indicates energy conversion, often showing a loss to other energy types, meaning the collision is inelastic.
This understanding is vital for physics students as they solve problems and apply these concepts to real-life scenarios, such as determining the effectiveness of safety mechanisms like airbags that redistribute kinetic energy during crashes.