Question

can u tell the formula for velocity of a particle on head on collision with other particle

Answer 1

The formulas for the velocities of the particles after a head-on collision are:

$v_1 = \frac{(m_1 - e m_2) u_1 + m_2 (1 + e) u_2}{m_1 + m_2}$

$v_2 = \frac{m_1 (1 + e) u_1 + (m_2 - e m_1) u_2}{m_1 + m_2}$

Where:

• $m_1, m_2$: masses of the particles
• $u_1, u_2$: initial velocities of the particles
• $v_1, v_2$: final velocities of the particles
• $e$: coefficient of restitution

Answer 2

To determine the velocities of particles after a head-on collision, we use two fundamental principles: the conservation of linear momentum and the definition of the coefficient of restitution.

Let:

• $m_1$ and $m_2$ be the masses of the first and second particles, respectively.
• $u_1$ and $u_2$ be their initial velocities before the collision.
• $v_1$ and $v_2$ be their final velocities after the collision.
• $e$ be the coefficient of restitution, which depends on the nature of the collision.

The two governing equations for a head-on collision are:

1. Conservation of Linear Momentum: The total momentum of the system before the collision is equal to the total momentum after the collision.

   $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$

2. Coefficient of Restitution (e): This relates the relative velocity of separation to the relative velocity of approach.

   $e = \frac{v_2 - v_1}{u_1 - u_2} \implies v_2 - v_1 = e (u_1 - u_2)$

Solving these two equations simultaneously for $v_1$ and $v_2$, we obtain the general formulas for the velocities of the particles after a head-on collision:

Velocity of the first particle ($v_1$):

$v_1 = \frac{(m_1 - e m_2) u_1 + m_2 (1 + e) u_2}{m_1 + m_2}$

Velocity of the second particle ($v_2$):

$v_2 = \frac{m_1 (1 + e) u_1 + (m_2 - e m_1) u_2}{m_1 + m_2}$

Special Cases for Coefficient of Restitution (e):

• Perfectly Elastic Collision: $e = 1$ (Kinetic energy is conserved).

  • If $m_1 = m_2$ and $u_2 = 0$, then $v_1 = 0$ and $v_2 = u_1$ (velocities interchange).

• Perfectly Inelastic Collision: $e = 0$ (Particles stick together after collision, moving with a common velocity).

  $v_1 = v_2 = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$

• Inelastic Collision: $0 < e < 1$ (Some kinetic energy is lost).

The formulas for the velocities of particles after a head-on collision are derived by simultaneously solving the equations for the conservation of linear momentum and the definition of the coefficient of restitution. These two principles are fundamental to analyzing collision dynamics. The coefficient of restitution, $e$, quantifies the elasticity of the collision, ranging from 0 for perfectly inelastic collisions (where particles stick together) to 1 for perfectly elastic collisions (where kinetic energy is conserved).