Conservation of linear momentum during impact

Conservation of linear momentum during impact:

If two bodies of mass m1 and m2 collide in air, the net external force acting on the system of bodies (m1+m2) is equal to
$ \displaystyle \vec{F_1} + \vec{m_1g} + \vec{F_2} + \vec{m_2 g} $

$ \displaystyle \vec{F_{net}} = \vec{F_1} + \vec{F_2} + \vec{m_1g} + \vec{m_2 g} $

We know that, during collision the impact forces F1 & F2 are equal in magnitude and opposite in direction. According to Newton’s 3rd law of motion,

$ \displaystyle \vec{F_1} + \vec{F_2} = 0 $

Now , we obtain,

$ \displaystyle \vec{F_{net}} = \vec{m_1g} + \vec{m_2 g} $

Since the impact takes place for very short time dt, the impulse of the force Fnet is

$ \displaystyle \vec{F_{net}}dt = (\vec{m_1g} + \vec{m_2 g})dt $

⇒ Impulse = $ \displaystyle (m_1 + m_2 )\vec{g}dt $

Since dt is a very small time interval, the impulse F.dt will be negligibly small. Because, impulse is equal to change in momentum of the system, a negligible impulse means negligible change of momentum. Let the change of momentum of 1 & 2 be ΔP1 & P2 respectively. The total change in momentum of the system

$ \displaystyle \vec{\Delta P} = \vec{\Delta P_1} + \vec{\Delta P_2} = \vec{F_{net}}dt = 0 $

$ \displaystyle \Delta(\vec{ P_1} + \vec{ P_2}) = 0 $

$ \displaystyle (\vec{ P_1} + \vec{ P_2}) = constant $

Therefore the net or total momentum of the colliding bodies remains practically unchanged along the line of action (impact) during the collision; In the other words, we can say that momentum of the system remains constant or conserved during the period of impact.
Therefore we can conveniently equate the net momentum of the colliding bodies at the beginning and at the end of the collision (or just before and just after the impact).

Note:

Remember that the impact force F is not an external force for the system of colliding bodies.

If no external force acts on the system, its momentum remains constant for all the times including the time of collision. Even if some external forces like gravitation, friction (known as non-impulsive forces in general) are present, we can conserve the momentum of the system during the impact, because the finite external forces cannot change the momentum of the system significantly in very short time.

Therefore, the change in position of the system during infinitesimal time of impact can also neglected.

Graphically, we observe that:

(a) The force becomes very large during a short time δt (time of collision).

(b) The external forces acting on the system are negligible compared to the impulsive force F acting during the time δt .

(c) The area of F-t curve during the time δt = $ \displaystyle \int_{0}^{\delta t}F dt $
It measures the impulse of the force during time δt or change of momentum of the system during that time.

Also Read :

Solved Problems based on Impulse
Analysis of Collision & Coefficient of restitution
Line of impact during collision
Head on Collision , Velocities after Collision
oblique impact in Collision

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