JEE Advance - Physics (2008 - Paper 1 Offline - No. 7)
Two balls, having linear momenta $${\overrightarrow p _1} = p\widehat i$$ and $${\overrightarrow p _2} = - p\widehat i$$, undergo a collision in free space. There is no external force acting on the balls. Let $${\overrightarrow {p'} _1}$$ and $${\overrightarrow {p'} _2}$$ 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 $${c_2}$$ :
$${\overrightarrow {p'} _1} = {a_1}\widehat i + {b_1}\widehat j + {c_1}\widehat k;{\overrightarrow {p'} _2} = {a_2}\widehat i + {b_2}\widehat j$$
$${\overrightarrow {p'} _1} = {c_1}\widehat k;{\overrightarrow {p'} _2} = {c_2}\widehat k$$
$${\overrightarrow {p'} _1} = {a_1}\widehat i + {b_1}\widehat j + {c_1}\widehat k;{\overrightarrow {p'} _2} = {a_2}\widehat i + {b_2}\widehat j - {c_1}\widehat k$$
$${\overrightarrow {p'} _1} = {a_1}\widehat i + {b_1}\widehat j;{\overrightarrow {p'} _2} = {a_2}\widehat i + {b_1}\widehat j$$
Explanation
In free space, there is no external force. Hence linear momentum of the system is conserved. Initial and final linear momentum of the system are
$$ \begin{aligned} & \vec{p}_i=\vec{p}_1+\vec{p}_2=\hat{p}-\hat{\imath} \hat{\imath}=\overrightarrow{0}, \\\\ & \vec{p}_f=\vec{p}_1^{\prime}+\vec{p}_2^{\prime} . \end{aligned} $$
The conservation of linear momentum, $\vec{p}_i=\vec{p}_f$ gives
$$ \vec{p}_1^{\prime}+\vec{p}_2^{\prime}=\overrightarrow{0} . $$
In case $(A)$,
$$ \begin{aligned} \vec{p}_1^{\prime}+\vec{p}_2^{\prime} & =\left(a_1+a_2\right) \hat{\imath}+\left(b_1+b_2\right) \hat{\jmath}+c_1 \hat{k} \\\\ & \neq \overrightarrow{0}, \quad\left(\because c_1 \neq 0\right) . \end{aligned} $$
In case $(B)$,
$$ \begin{aligned} \vec{p}_1^{\prime}+\vec{p}_2^{\prime} & =\left(c_1+c_2\right) \hat{k} \\\\ & =\overrightarrow{0}, \quad\left(\text { if } c_2=-c_1\right) . \end{aligned} $$
In case $(\mathrm{C})$,
$$ \begin{aligned} \vec{p}_1^{\prime}+\vec{p}_2^{\prime} & =\left(a_1+a_2\right) \hat{\imath}+\left(b_1+b_2\right) \hat{\jmath} \\\\ & =\overrightarrow{0}, \quad\left(\text { if } a_2=-a_1 \text { and } b_2=-b_1\right) . \end{aligned} $$
In case (D),
$$ \begin{aligned} \vec{p}_1^{\prime}+\vec{p}_2^{\prime} & =\left(a_1+a_2\right) \hat{\imath}+2 b_1 \hat{\jmath} \\\\ & \neq \overrightarrow{0}, \quad\left(\because b_1 \neq 0\right) . \end{aligned} $$
$$ \begin{aligned} & \vec{p}_i=\vec{p}_1+\vec{p}_2=\hat{p}-\hat{\imath} \hat{\imath}=\overrightarrow{0}, \\\\ & \vec{p}_f=\vec{p}_1^{\prime}+\vec{p}_2^{\prime} . \end{aligned} $$
The conservation of linear momentum, $\vec{p}_i=\vec{p}_f$ gives
$$ \vec{p}_1^{\prime}+\vec{p}_2^{\prime}=\overrightarrow{0} . $$
In case $(A)$,
$$ \begin{aligned} \vec{p}_1^{\prime}+\vec{p}_2^{\prime} & =\left(a_1+a_2\right) \hat{\imath}+\left(b_1+b_2\right) \hat{\jmath}+c_1 \hat{k} \\\\ & \neq \overrightarrow{0}, \quad\left(\because c_1 \neq 0\right) . \end{aligned} $$
In case $(B)$,
$$ \begin{aligned} \vec{p}_1^{\prime}+\vec{p}_2^{\prime} & =\left(c_1+c_2\right) \hat{k} \\\\ & =\overrightarrow{0}, \quad\left(\text { if } c_2=-c_1\right) . \end{aligned} $$
In case $(\mathrm{C})$,
$$ \begin{aligned} \vec{p}_1^{\prime}+\vec{p}_2^{\prime} & =\left(a_1+a_2\right) \hat{\imath}+\left(b_1+b_2\right) \hat{\jmath} \\\\ & =\overrightarrow{0}, \quad\left(\text { if } a_2=-a_1 \text { and } b_2=-b_1\right) . \end{aligned} $$
In case (D),
$$ \begin{aligned} \vec{p}_1^{\prime}+\vec{p}_2^{\prime} & =\left(a_1+a_2\right) \hat{\imath}+2 b_1 \hat{\jmath} \\\\ & \neq \overrightarrow{0}, \quad\left(\because b_1 \neq 0\right) . \end{aligned} $$
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