Question

Prove m1u1+m2u2=m1v1+m2v2

Answer 1

Explanation:

Derivation of Conservation of Momentum

Let us consider a situation wherein: a truck of mass m1, velocity u1 and its momentum = m1u1 and a car of mass m2, velocity u2 and its momentum m2u2; are moving in the same direction but with different speeds. Therefore, total momentum=m1u1 + m2u2.

Now suppose the car and truck collide for a short time t, their velocities will change. So now the velocity of the truck and car become v1 and v2 respectively. However, their mass remains the same. Hence, now the total momentum = m1v1 + m2v2

Acceleration of car (a) = (v2–u2)/t

Also, F = ma

F1 = Force exerted by truck on the car

F1 = m2(v2–u2)/t

Acceleration of truck =(v1–u1)/t

F2 = m1(v1–u1)/t and F1 = –F2

m2(v2– u2)/t = –m1(v1– u1)/t

m2v2–m2u2 = –m1v1+m1u1

or m1u1+m2u2 = m2v2+m1v1

Answer 2

m₁u₁+m₂u₂=m₁v₁+m₂v₂ states that the total momentum before the collision is equal to the total momentum after the collision.

Given :

m₁u₁+m₂u₂=m₁v₁+m₂v₂

To find :

Prove that m₁u₁+m₂u₂=m₁v₁+m₂v₂

Solution :

Let m₁ and m₂ be the masses of two bodies A and B respectively

Let u₁ and u₂ be the initial velocities of A and B

And v₁, v₂ be the final velocities.

According to Newton's third law,

                                          F₁ = - F₂

                                       m₁a₁ = - m₂a₂

                       m₁ ( v₁ - u₁ / t )  =  m₂ ( v₂ - u₂ / t )

                               m₁ (v₁ - u₁) = m₂ (v₂ - u₂)

                              m₁v₁ - m₁u₁ = m₂v₂ - m₂u₂

                            m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Hence proved.

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