Question

State and prove the law of conservation of linear momentum.

Answer 1

1. There is no change in the linear momentum of a system of bodies as long as no net external force acts on them.
2. Let us prove the law of conservation of linear momentum with the following illustration:
   
3. Let two bodies A and B having masses m1 and m2 move with initial velocity u₁ and u₂ in a straight line.
4. Let the velocity of the first body be higher than that of the second body. i.e., u₁ > u₂.
5. During an interval of time t second, they tend to have a collision. After the impact, both of them move along the same straight line with a velocity v₁ and v₂ respectively.
   Force on body B due to A,
   F_B = `("m"_2["v"_2-"u"_2])/"t"`
   Force on body A due to B,
   F_A = `("m"_1["v"_1-"u"_1])/"t"`
   By Newton’s III law of motion, Action force = Reaction force
   F_A = -F_{B
   `("m"_1["v"_1-"u"_1])/"t"` = `("m"_2["v"_2-"u"_2])/"t"`}

m₁v₁ + m₂v₂ = m₁u₁ + m₂u₂
The above equation confirms in the absence of an external force, the algebraic sum of the momentum after collision is numerically equal to the algebraic sum of the momentum before collision.
Hence the law of conservation linear momentum is proved.