A) \[\frac{\sqrt{3}}{2}v\]
B) \[\sqrt{3}v\]
C) \[\frac{1}{2}v\]
D) \[\frac{v}{\sqrt{3}}\]
Correct Answer: A
Solution :
The horizontal component of velocity \[{{v}_{x}}=v\,\cos {{45}^{o}}\] \[=\frac{v}{\sqrt{2}}\] Vertical component of velocity \[{{v}_{y}}=v\text{ }\sin \text{ }45{}^\circ \] \[=\frac{v}{\sqrt{2}}\] After collision \[{{v}_{x}}=\frac{v}{\sqrt{2}}\] Now, coefficient of restitution is \[e=\frac{normal\text{ }component\text{ }of\text{ }velocity\text{ }of\text{ }separation}{normal\text{ }component\text{ }of\text{ }velocity\,of\text{ }approach}\] \[=\frac{v{{'}_{y}}-0}{{{v}_{y}}-0}\] \[\Rightarrow \]\[v{{'}_{y}}=e{{v}_{y}}=\frac{1}{\sqrt{2}}\times \frac{v}{\sqrt{2}}=\frac{v}{2}\] Thus, velocity of reflection \[v'=\sqrt{{{(v_{x}^{'})}^{2}}+{{(v_{y}^{'})}^{2}}}\] \[=\sqrt{{{\left( \frac{v}{\sqrt{2}} \right)}^{2}}+{{\left( \frac{v}{2} \right)}^{2}}}\] \[=\sqrt{{{\left( \frac{3{{v}^{2}}}{4} \right)}^{2}}}=\frac{\sqrt{3}}{2}v\]
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