Question

A particle of mass m is moving with speed v collides elastically with another particle of mass 2m. Find the speed of smaller mass after head-on collision? A) V/3 B) -(V/3) C) 2V/3 D) -(2V/3)

Answer 1

Answer :

◈ We know that, coefficient of restitution for elastic collision = 1

➝ e = (v₂ - v₁) / (u₁ - u₂)

➝ e(u₁ - u₂) = (v₂ - v₁)

➝ 1(v - 0) = v₂ - v₁

➝ v₂ = v + v₁ ...... (I)

◈ Linear momentum is conserved during any kind of collision$.$

➝ m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

➝ mv + 2m(0) = mv₁ + 3mv₂

➝ mv = mv₁ + 2m(v + v₁) [Eq. (I)]

➝ v = v₁ + 2v + 2v₁

➝ v - 2v = 3v₁

➝ v₁ = -v/3

Hence, option (B) is the correct answer!

Answer 2

Given :

• A particle of mass m is moving with speed v
• Another mass 2m , which is intallty at rest position
• Collision between m and 2m is elastic

To find :

The speed of smaller mass after head-on collision .

Theory :

• Cofficient of Restitution :

$\sf\:e=\dfrac{V\:separation\:along\:line\:of\:contact}{V\:approach\:along\:line\:of\:impact}$

$\bf\implies\:e=\dfrac{v_2-v_1}{u_1-u_2}$

•Collision types :

1) Perfectly elastic collision

• In addition to Momentum , K.E also be conserved.
• e=1

2) Inelastic collision

• K.E after collision < K.E before collision
• 0<e<1

3) Perfectly elastic collision

• Maximum loss of K.E
• e=0

Solution :

We know that :

$\sf\:e=\dfrac{v_2-v_1}{u_1-u_2}$

Since , collision is elastic , thus e= 1

and 2m mass is initially at rest.

$\sf\implies\:1=\dfrac{v_2-v_1}{u_1-0}$

here $\sf\:u_1=v$ (given )

$\sf\implies\:v=v_2-v_1....(1)$

Since , there is no $\sf\:F_{ext}$ force then ,

$\sf\:P_i=P_f$

$\sf\:m_1u_1+m_2u_2=m_1v_1+m_2v_2$

$\sf\implies\:mv+m\times0=mv_1+2mv_2$

$\sf\implies\:mv=mv_1+2mv_2$

$\sf\implies\:mv=m(v_1+2v_2)$

$\sf\implies\:v=(v_1+2v_2)...(2)$

Now , Multiply equation (1) by 2 then,

$\sf\:2v=2v_2-2v_1...(3)$

Now subtract equation (1) form (3) then ,

$\sf\:2v-v=2v_2-2v_1-(v_1+2v_2)$

$\sf\:v=2v_2-2v_1-v_1-2v_2$

$\sf\:v=-3v_1$

$\sf\:v_1=\dfrac{-v}{3}$

Therefore, The speed of smaller mass after head-on collision is -v/3

Correct option b)

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More About the topic :

Collision :

It is a phenomenon in which very large force acts for a very short interval of time.