A) \[\frac{\text{u+v}}{2}\]
B) \[\frac{1}{2}\sqrt{{{\text{u}}^{\text{2}}}\text{+}{{\text{v}}^{\text{2}}}}\]
C) \[\sqrt{\text{uv}}\]
D) \[\sqrt{\left( \frac{{{\text{u}}^{2}}+{{\text{v}}^{2}}}{2} \right)}\]
Correct Answer: D
Solution :
Let 'S' be the distance between two ends 'a' be the constant acceleration. As we know \[{{\text{v}}^{\text{2}}}-{{\text{u}}^{\text{2}}}\text{=2aS or, aS=}\frac{{{\text{v}}^{\text{2}}}-{{\text{u}}^{\text{2}}}}{2}\]. Let v be velocity at mid-point. Therefore, \[\text{v}_{\text{c}}^{\text{2}}-{{\text{u}}^{\text{2}}}\text{=2a}\frac{\text{S}}{\text{2}}\Rightarrow \text{v}_{\text{c}}^{\text{2}}\text{=}{{\text{u}}^{\text{2}}}\text{+aS}\] \[\text{v}_{\text{c}}^{\text{2}}={{\text{u}}^{\text{2}}}+\frac{{{\text{v}}^{2}}-{{\text{u}}^{\text{2}}}}{2}\Rightarrow {{\text{v}}_{\text{c}}}=\sqrt{\frac{{{\text{u}}_{\text{2}}}\text{+}{{\text{v}}^{\text{2}}}}{\text{2}}}\]S
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