Two balls, having linear momenta $\vec{p}_{1}=p \hat{i}$ and $\vec{p}_{2}=-p \hat{i}$, undergo a collision in free space. There is no external force acting on the balls. Let $\vec{p}_{1}^{\prime}$ and $\vec{p}_{2}^{\prime}$ be their final momenta. The following option(s) is(are) NOT ALLOWED for any non-zero value of $p, a_{1}, a_{2}, b_{1}, b_{2}, c_{1}$ and $c_{2}$.

(A) $\vec{p}_{1}^{\prime}=a_{1} \hat{i}+b_{1} \hat{j}+c_{1} \hat{k}$ $\vec{p}_{2}^{\prime}=a_{2} \hat{i}+b_{2} \hat{j}$; (D) $\vec{p}_{1}^{\prime}=a_{1} \hat{i}+b_{1} \hat{j}$ $\vec{p}_{2}^{\prime}=a_{2} \hat{i}+b_{1} \hat{j}$

(B) $\vec{p}_{1}^{\prime}=c_{1} \hat{k}$ $\vec{p}_{2}^{\prime}=c_{2} \hat{k}$

(C) $\vec{p}_{1}^{\prime}=a_{1} \hat{i}+b_{1} \hat{j}+c_{1} \hat{k}$ $\vec{p}_{2}^{\prime}=a_{2} \hat{i}+b_{2} \hat{j}-c_{1} \hat{k}$

JEE Advanced · 2008 · 13 Apr 2008 · Paper 1 · Conservation of Momentum

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Two balls, having linear momenta

p_{1} = p \hat{i}

and

p_{2} = - p \hat{i}

, undergo a collision in free space. There is no external force acting on the balls. Let

p_{1}^{'}

and

p_{2}^{'}

be their final momenta. The following option(s) is(are) NOT ALLOWED for any non-zero value of

p, a_{1}, a_{2}, b_{1}, b_{2}, c_{1}

and

c_{2}

(A) $p_{1}^{'} = a_{1} \hat{i} + b_{1} \hat{j} + c_{1} \hat{k}$
$p_{2}^{'} = a_{2} \hat{i} + b_{2} \hat{j}$ ✓ Correct
(B) $p_{1}^{'} = c_{1} \hat{k}$
$p_{2}^{'} = c_{2} \hat{k}$
(C) $p_{1}^{'} = a_{1} \hat{i} + b_{1} \hat{j} + c_{1} \hat{k}$
$p_{2}^{'} = a_{2} \hat{i} + b_{2} \hat{j} - c_{1} \hat{k}$
(D) $p_{1}^{'} = a_{1} \hat{i} + b_{1} \hat{j}$
$p_{2}^{'} = a_{2} \hat{i} + b_{1} \hat{j}$ ✓ Correct

Answer: (A), (D)

Solution

Since there is no external force, momentum is conserved:

p_{1} + p_{2} = p_{1}^{'} + p_{2}^{'}

. Initially,

p_{1} + p_{2} = p \hat{i} - p \hat{i} = 0

. Thus

p_{1}^{'} + p_{2}^{'} = 0

, meaning the final momenta must be equal and opposite. Option (A) violates this because the

\hat{k}

components don't cancel (

c_{1} + 0 \neq = 0

). Option (D) violates this because the

\hat{j}

components are equal (

b_{1} + b_{1} \neq = 0

). Options (B) and (C) satisfy momentum conservation.

JEE Advanced · 2008 · 13 Apr 2008 · Paper 1 · One or more correct options

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JEE Advanced · 2008 · 13 Apr 2008 · Paper 1 · Conservation of Momentum

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