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Chapter 10: Problem 17

Some inequalities in elastic collisions Use the elastic scattering formulae to show the following inequalities: (i) When \(m_{1}>m_{2}\), the scattering angle \(\theta_{1}\) is restricted to the range \(0 \leq \theta_{1} \leq \sin ^{-1}\left(m_{2} / m_{1}\right)\). (ii) If \(m_{1}m_{2}\), the opening angle is acute. (iii) $$ \frac{E_{1}}{E_{0}} \geq\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2}, \quad \frac{E_{2}}{E_{0}} \leq \frac{4 m_{1} m_{2}}{\left(m_{1}+m_{2}\right)^{2}} $$.

Short Answer

Expert verified
The inequality for the scattering angle comes from the properties of the elastic collisions and the relation between angles and the masses of the colliding objects. The second inequality, about the opening being obtuse or acute, is related to how the collision's angle changes based on the relative masses of the objects. Lastly, the inequality about the kinetic energy results from the principle of the conservation of kinetic energy in elastic collisions.

Step by step solution

01

Analyzing the inequalities for the scattering angle

This exercise involves demonstrating that the scattering angle of a collision, \( \theta_{1} \), is restricted by the relative masses of the two colliding objects. It is assumed that \( \theta_{1} \) cannot be less than 0 or greater than \( \sin^{-1}(m_{2} / m_{1}) \) when \( m_{1} > m_{2} \). Hence the first inequality occurs.
02

Demonstrating the acute and obtuse angles

If \( m_{1} < m_{2} \), then the opening angle is said to be obtuse. Conversely, if \( m_{1} > m_{2} \), the angle is acute. This happens because with a bit larger mass, the object will tend to keep its direction while the other object with less mass will deviate more, forming an acute angle if \( m_{1} > m_{2} \), otherwise forming an obtuse angle when \( m_{1} < m_{2} \). This is how the second inequality arises.
03

Calculating the kinetic energy preservation

The third part of the exercise involves kinetic energy. According to the principle of conservation of kinetic energy in elastic collisions, the total kinetic energy before and after the collision is conserved. Thus, the following inequalities hold after simplifying the expressions: \( \frac{E_{1}}{E_{0}} \geq \left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2} \), \( \frac{E_{2}}{E_{0}} \leq \frac{4 m_{1} m_{2}}{\left(m_{1}+m_{2}\right)^{2}} \). This part of the inequality is derived from the conservation of momentum and kinetic energy imposed by the collision.

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Key Concepts

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

Scattering Angle
The scattering angle, denoted as \( \theta \), is a crucial parameter in the analysis of elastic collisions. It measures the direction of the velocity vector of a particle after collision with respect to its original direction. Imagine two billiard balls striking each other; the scattering angle would describe the direction one of the balls takes relative to its original path after they collide.

For elastic collisions, when one object with mass \( m_1 \) collides with another object with mass \( m_2 \), the scattering angle of the first object \( \theta_1 \) depends on the relative masses. If \( m_1 > m_2 \), the larger mass will not deflect as much, and therefore \( \theta_1 \) is restricted to smaller values, specifically between 0 and \( \sin^{-1}(\frac{m_2}{m_1}) \). Understanding this concept is key to solving problems involving direction after a collision and is rooted in the conservation of momentum.
Conservation of Kinetic Energy
In an elastic collision, the total kinetic energy of the system is conserved. This means that the kinetic energy before and after the impact remains constant. For instance, in an isolated system where two cars collide and bounce off each other without any external forces or energy losses, the sum of their kinetic energies remains unchanged.

The formula \( \frac{E_1}{E_0} \geq \left(\frac{m_1 - m_2}{m_1 + m_2}\right)^2 \), where \( E_1 \) is the kinetic energy of the first object after the collision and \( E_0 \) is its initial kinetic energy, is a manifestation of this principle. The inequality indicates that while kinetic energy is conserved as a whole, it might be redistributed in individual bodies, depending on their masses. This concept is a cornerstone in understanding any type of elastic collision.
Elastic Scattering Formulae
Elastic scattering formulae enable calculations regarding how objects scatter after an elastic collision. These formulae are essential for predicting outcomes in physics, engineering, and even sports science. They describe not just the velocities of the scattering objects but also the angles at which they deviate from their original paths.

For example, the formula \( \frac{E_2}{E_0} \leq \frac{4 m_1 m_2}{(m_1 + m_2)^2} \) relates to the kinetic energy \( E_2 \) of the second object after the collision, compared to the initial kinetic energy \( E_0 \) of the first object. The forms of these inequalities provide insight into how kinetic energy is distributed as a result of the masses involved and the conservation laws that govern the collision.
Momentum Conservation
Momentum conservation is a fundamental principle in physics which states that the total momentum of a closed system is constant if no external forces act upon it. This principle applies to all sorts of collisions, including elastic ones. In the context of elastic collisions, the momentum vector's magnitude and direction are conserved separately.

The transfer of momentum from one object to another in a collision can be complex, but the total momentum before and after the event remains the same. For example, when a moving cue ball strikes a stationary eight ball, the momentum is transferred, and both balls move after impact. However, the combined momentum of the balls post-collision will equal the cue ball's momentum before the strike. Calculations often use the formulae for conservation of momentum along with those for conservation of kinetic energy to solve collision problems.

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Most popular questions from this chapter

Two particles with masses \(m_{1}, m_{2}\) and velocities \(v_{1}, v_{2}\) collide and stick together. Find the velocity of this composite particle and show that the loss in kinetic energy due to the collision is $$ \frac{m_{1} m_{2}}{2\left(m_{1}+m_{2}\right)}\left|v_{1}-v_{2}\right|^{2} $$.

A boat of mass \(M\) is at rest in still water and a man of mass \(m\) is sitting at the bow. The man stands up, walks to the stem of the boat and then sits down again. If the water offers a resistance to the motion of the boat proportional to the velocity of the boat, show that the boat will eventually come to rest at its orginal position. [This remarkable result is independent of the resistance constant and the details of the man's motion.]

Show that, if a system moves periodically, then the average of the total external force over a period of the motion must be zero. A juggler juggles four balls of masses \(M, 2 M, 3 M\) and \(4 M\) in a periodic manner. Find the time average (over a period) of the total force he applies to the balls. The juggler wishes to cross a shaky bridge that cannot support the combined weight of the juggler and his balls. Would it help if he juggles his balls while he crosses?

A body of mass \(4 m\) is at rest when it explodes into three fragments of masses \(2 m, m\) and \(m\). After the explosion the two fragments of mass \(m\) are observed to be moving with the same speed in directions making \(120^{\circ}\) with each other. Find the proportion of the total kinetic energy carried by each fragment.

Comparison with quantum scattering A uniform flux of particles is incident upon a fixed hard sphere of radius \(a\). The particles that strike the sphere are reflected elastically. Show that the differential scattering cross section is \(\sigma(\theta)=a^{2} / 4\) and that the total cross section is \(S=\pi a^{2}\) The solution of the same problem given by quantum mechanics is $$ \sigma(\theta)=\frac{a^{2}}{(k a)^{2}}\left|\sum_{l=0}^{\infty} \frac{(2 l+1) j_{l}(k a) P_{l}(\cos \theta)}{h_{l}(k a)}\right|^{2}, \quad S=\frac{4 \pi a^{2}}{(k a)^{2}} \sum_{l=0}^{\infty}\left|\frac{(2 l+1) j_{l}(k a)}{h_{I}(k a)}\right|^{2} $$ where \(P_{l}(z)\) is the Legendre polynomial of degree \(l\), and \(j_{l}(z), h_{I}(z)\) are spherical Bessel functions order \(l\). (Stay cool: these special functions should be available on your computer package.) The parameter \(k\) is related to the particle momentum \(p\) by the formula \(p=\hbar k\), where \(\hbar\) is the modified Planck constant. When \(k a\) is large, one would expect the quantum mechanical values for \(\sigma(\theta)\) and \(S\) to approach the classical values. Calculate the quantum mechanical values numerically for \(k a\) up to about 30 (the calculation becomes increasingly difficult as \(k a\) increases), using about 100 terms of the series. The author's results are shown in Figure 10.11. The quantum mechanical value for \(\sigma(\theta)\) does approach the classical value for larger scattering angles, but behaves very erratically for small scattering angles. Also, the value of \(S\) tends to twice the value expected! Your physics lecturer will be pleased to explain these interesting anomalies.
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