One-dimensional Collision

One-dimensional Collision

When the children play marbles, then if one marble collides with another marble and after the collision if the marbles move in a straight line, then that collision is called one-dimensional collision. That means, if the relative motion of the two colliding bode is along the same straight line below and after the collision, then that collision is called one dimensional collision.

Suppose two particles of masses m₁ and m₂ are moving in a straight line along x-axis with velocities v₀₁ and v₀₂ respectively (Figure]. Here v₀₁ > v₀₂. At one time the first body hits the second body from behind and afterwards the two particles move along the same straight line with velocities v₁ and v₂ respectively.

Here action force applied on the body of mass m₂ is F₁ and the mass m₂ also creates reaction force F₂ on m₁.

Again, suppose if the duration of the action and reaction is t,

then sum of initial momentum of the two particles = m₁v₀₁ + m₂v₀₂ … … … (1)

and the sum of the final momentum of the two particles = m₁v₁ + m₂v₂

According to Newton’s third law of motion,

action = reaction

F₂ = – F₁

During collision the action and reaction forces act for the same duration.

Suppose, the two bodies move along the same straight line, before and after the collision, with velocities v₀₁ and v₀₂ respectively. Due to action and reaction, accelerations of the two bodies become a₁ and a₂.

then, F₁ = – F₂

or, m₁a₁ = – m₂a₂

or, m₁ [(v₁ – v₀₁) / t] = – m₂ [(v₂ – v₀₂) / t]

or, m₁v₁ – m₁v₀₁ = – m₂v₂ + m₂v₀₂

According to the principle of conservation of momentum,

Momentum before collision = Momentum after collision.

This is the equation of one-dimensional collision.