# State and prove the law of conservation of linear momentum.

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State and prove the law of conservation of linear momentum.

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According to the law of conservation of momentum

When two or more bodies act upon each other their total momentum remains constant provided no external forces are acting

•So, Momentum is never created or destroyed.

•When this law is applied for a collision between two bodies, the total momentum of the colliding bodies before the collision is equal to the total momentum after the collision.

•We can apply this law for a collision between two vehicles. This law is applicable for all types of collisions.

•Consider two particles say A and B of mass m1 and m2 collide with each other and forces acting on these particles are only the ones they exert on each other.

•Let u1 and v1 be the initial and final velocities of particle A and similarly, u2 and vfor particle B. Let the two particles be in contact for a time t.

So, Change in momentum of A = m1 (v1-u1) Change in the momentum of B = m2 (v2-u2

•During the collision, let A impart an average force equal to FBA on B and let B exert an average FAB on A.

We know that from third law of motion FBA = FAB eq. (4)

FBA = m2 x a2$\frac{m_2(v_2-u_2)}{t}$

FAB = m1 x a1 = $\frac{m_1(v_1-u_1)}{t}$

Canceling t on both sides and rearranging the equation we get m1 u1 + m2 u2 = m1 v1 + m2 v2 eq.(5)

Now, m1 u1 + m2 u2 represents the total momentum of particles A and B before collision and m1 v1 + m2 v2 represents the total momentum of particles after the collision. This means that

Total momentum before collision = total momentum after the collision

•Equation 5 which m1 u1 + m2 u2 = m1 v1 + m2 v2, is known as the law of conservation of momentum.