Osm Physics by Ashish

Head-On Inelastic Collision of Two Particles (b) Head-On Inelastic Collision of Two Particles Which Stick Together Consider two particles of masses \( m_1 \) and \( m_2 \). The first particle is moving with velocity \( u_1 \), while the second particle is initially at rest. They collide head-on and, after collision, stick together and move with a common velocity \( v \). (i) In the Laboratory Frame of Reference (L-Frame) Let the initial velocity of particle \( m_1 \) be \( u_1 \), and that of particle \( m_2 \) be zero. According to the law of conservation of linear momentum, \[ m_1 u_1 + m_2(0) = (m_1 + m_2)v \] Therefore, \[ \boxed{v = \dfrac{m_1 u_1}{m_1 + m_2}} \tag{1} \] Loss of Kinetic Energy The kinetic energy before collision is \[ K_i = \dfrac{1}{2} m_1 u_1^2 \] and the kinetic energy after collision is \[ K_f = \dfrac{1}{2}(m_1 + m_2)v^2. \] Using Eq...

Consider a particle of mass m₁, moving with a constant velocity u₁, which collides elastically with a stationary particle of mass m₂ in the laboratory frame of reference. After the collision, the particle of mass m₁ moves with velocity v₁, making an angle θ₁ with its initial direction of motion, while the particle of mass m₂ moves with velocity v₂, making an angle θ₂ with the initial direction of motion of the first particle. Let the initial direction of motion of m₁ be along the X-axis, and let the velocities v₁ and v₂ lie in the X–Y plane, as shown in Fig. 1. Fig. 1: Elastic collision in two dimensions in Lab frame   The velocity of the centre of mass in the laboratory frame is V CM = m₁u₁ / (m₁ + m₂)              (1) The initial velocity of particle m₁ in the C.M. frame is u₁′ = u₁ − V_C   (2) Using Eq. (1), u₁′ = m₂u₁ / (m₁ + m₂)          ...

  If heat is transferred from system A to system A′ (the surroundings) and the temperature of A′ remains unchanged, then system A′ is called a heat reservoir. Let system A be in its i-th state with energy Eᵢ. The combined system A*, consisting of A and A′, is completely isolated. Therefore, its total energy E* remains constant. Hence, when the energy of system A is Eᵢ, the energy of the reservoir A′ is E′ = E* − Eᵢ.    (1) According to the principle of a priori probabilities, all accessible microscopic states of an isolated system are equally probable. Therefore, the probability Pᵢ of finding system A in its specific i-th state is directly proportional to the number of accessible states of the reservoir: Pᵢ ∝ Ω′(E′). Thus, Pᵢ = C′Ω′(E′), or, using Eq. (1), Pᵢ = C′Ω′(E* − Eᵢ).    (2) where C′ is a proportionality constant independent of the i-th state. Since system A is very small compared with the heat reservoir A′, Eᵢ ≪ E*, and therefore E′ ≈ E*. The ...

Let the position vectors of two particles of masses and at any instant be and , respectively. The position vector of the centre of mass at that instant is The velocity of the centre of mass is therefore where and are the initial velocities of particles 1 and 2 in the laboratory frame. Initial velocities in the C-reference frame The initial velocity of the first particle in the C-reference frame is Similarly, the initial velocity of the second particle in the C-reference frame is Thus, total momentum before collision in the C-reference frame Therefore, in the absence of an external force, the total momentum before collision in the C-reference frame is zero . Hence, by the law of conservation of linear momentum, if and are the velocities after collision in the C-reference frame, Therefore, Application of conservation of kinetic energy Since the collision is elastic, the law of conservation of kinetic energy is also valid in the C-reference frame: ...