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Chapter 6: Problem 30

A microscopic particle collides with another stationary microscopic particle of the same mass. 'lhe angle of divergence is/are: (a) Equals \(\frac{\pi}{2}\), when the collision is ideally elastic. (b) Less than \(\frac{\pi}{2}\), when the collision is inelastic. (c) Always \(\frac{\pi}{2}\), independent of the nature of collision. (d) Always less than \(\frac{\pi}{4}\) and depends on the collision.

Short Answer

Expert verified
Option (a) is correct; in an elastic collision, the angle is \(\pi/2\).

Step by step solution

01

Understand Elastic Collision in Two Dimensions

In an elastic collision, both momentum and kinetic energy are conserved. For two particles with equal mass, when one particle is initially at rest, the paths after the collision will form angles with the original direction of the incident particle. This type of scatter results in an angle between the two diverging paths that must add up to \(\pi/2\) due to kinetic energy conservation.
02

Application to the Given Problem

For an ideally elastic collision, when the particles have equal mass and one is initially stationary, the angle of divergence between the paths of the two particles after the collision is exactly \(\pi/2\). This is because in such collisions, each path makes an angle \(\theta\) and \(\pi/2 - \theta\) with respect to the original direction, satisfying the condition.
03

Analyze Other Options for Inelastic Collision

In an inelastic collision, kinetic energy is not conserved, although momentum is still conserved. This change affects the angles formed between the paths after the collision. Typically, the angle of divergence will be less than \(\pi/2\) due to some of the kinetic energy being converted into other forms of energy, therefore option (b) is plausible.
04

Determine Which Option is Supported by Elastic Collision

Given that the elastic scenario specified results in an angle of \(\pi/2\), option (a) is correct. Options such as (c) contradict the special condition under elastic collision, and (d) provides no physical rationale within standard mechanics for angles less than \(\pi/4\).

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Key Concepts

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

Momentum Conservation
Momentum is a fundamental concept in physics, and it plays a crucial role during collisions. In any collision, whether elastic or inelastic, momentum is always conserved. This means that the total momentum of a system before a collision is equal to the total momentum after the collision.
  • Momentum is a vector quantity, which means it has both magnitude and direction.
  • It can be mathematically expressed as the product of mass and velocity, i.e., \( p = mv \).
  • In a collision involving two particles, the sum of their momenta before the collision will equal the sum of their momenta after the collision.
This principle allows us to apply the law of conservation of momentum to predict the final velocities and directions of particles post-collision. Essentially, no matter how the particles collide, the momentum remains shared and conserved among them.
Kinetic Energy Conservation
Kinetic energy conservation is a unique feature of elastic collisions. In an elastic collision, not only is momentum conserved, but the kinetic energy of the system is conserved too.
  • Kinetic energy is defined as \( KE = \frac{1}{2}mv^2 \).
  • This conservation implies that the sum of the kinetic energies of the particles before the collision is the same after they collide.
  • Elastic collisions are ideally described by the fact that no kinetic energy is transformed into other forms of energy, such as heat or sound.
Unlike momentum conservation, which holds true for all types of collisions, kinetic energy conservation is specific to elastic collisions. A good example is billiard balls striking one another or micromolecules interacting in a low-energy state.
Inelastic Collision
In contrast to elastic collisions, inelastic collisions are characterized by a loss of kinetic energy. While momentum remains conserved, some of the kinetic energy is transformed into other energy forms such as heat or deformation energy.
  • In a perfectly inelastic collision, particles might stick together, resulting in maximum loss of kinetic energy.
  • Most real-world collisions fall into the inelastic category, where only part of the kinetic energy is conserved.
  • This energy conversion impacts the outcome, such as the speed and direction of the objects after they collide.
Because kinetic energy is not conserved, the angle of divergence is typically less than \(\pi/2\), offering more varied possible outcomes after the collision.
Angles of Divergence
Angles of divergence are pivotal in understanding the outcome of a collision, particularly in microscopic or small scale events. For elastic collisions where two objects have equal mass, the sum of the angles between their paths post-collision is \(\pi/2\).
  • This is a direct consequence of kinetic energy conservation, dictating how energy is distributed between the objects.
  • In elastic collisions, each object deflects off the other, and the angles formed relative to the initial direction add up precisely to right angles.
  • When collisions are inelastic, the angle of divergence is typically less than \(\pi/2\) due to energy loss, influencing the scatter pattern.
Understanding these angles helps in predicting motion trajectories and analyzing different types of collisions based on observed angles. It highlights the delicate balance between forces and energy within collision dynamics.

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Most popular questions from this chapter

A ball of mass \(2 m\), moving due east with a speed \(8 u\) collides head on with a ball \(B\) of mass \(m\). The velocity of \(B\), just after the collision is \(5 u\) due cast. If the coefficient of restitution is \(1 / 11\), what percentage of the initial \(\mathrm{KE}\) is lost?

\(\Lambda\) ball moving with a specd of \(9 \mathrm{~m} / \mathrm{s}\) surikes an identical stationary ball such that aficr the collision the direction of cach ball makes an angle of \(30^{\circ}\) with the original line of motion. \Gammaind the specd of cach ball aficr the collision.

Two perfectly inelastic bodies of masses \(m_{1}\) and \(m_{2}\) moving with velocities \(u_{1}\) and \(u_{2}\) in the same direction impinge directly. Then select the correct alternative (a) There is no loss in kinetic energy (b) The loss in kinetic energy is \(\frac{m_{1} m_{2}}{2\left(m_{1}+m_{2}\right)}\left(u_{1} \quad u_{2}\right)^{2}\) (c) \(\Lambda\) fier the collision, they move with common velocity (d) There is no loss in momentum of the system

A block of mass \(2.0 \mathrm{~kg}\) moving at \(2.0 \mathrm{~m} / \mathrm{s}\) collides head on with another block of equal mass kept at rest. If the actual loss in kinetic energy is half of the maximum loss in kinetic energy, find the coefficient of restitution. (a) 2 (b) \(\frac{1}{2}\) (c) \(\sqrt{2}\) (d) \(\frac{1}{\sqrt{2}}\)

Locate the c.m. of a sector of uniform circular disc of radius \(R\) and of mass \(m\) and il substends an angle \(\theta_{0}\) at its cenire.
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