A) \[\rho V{{v}_{1}}\]
B) \[\rho V({{v}_{1}}+{{v}_{2}})\]
C) \[\frac{\rho V}{{{v}_{1}}+{{v}_{2}}}v_{1}^{2}\]
D) \[\rho \left[ \frac{V}{{{v}_{2}}} \right]{{({{v}_{1}}+{{v}_{2}})}^{2}}\]
Correct Answer: D
Solution :
Force acting on plate, \[F=\frac{dp}{dt}=v\ \left( \frac{dm}{dt} \right)\] Mass of water reaching the plate per sec =\[\frac{dm}{dt}\] \[=Av\rho =A({{v}_{1}}+{{v}_{2}})\rho \]\[=\frac{V}{{{v}_{2}}}({{v}_{1}}+{{v}_{2}})\rho \] (\[v={{v}_{1}}\,+\,{{v}_{2}}\,=\]velocity of water coming out of jet w.r.t. plate) (\[A=\] Area of cross section of jet \[=\frac{V}{{{v}_{2}}}\]) \[\therefore \] \[F=\frac{dm}{dt}v=\frac{V}{{{v}_{2}}}({{v}_{1}}+{{v}_{2}})\rho \times ({{v}_{1}}+{{v}_{2}})\] \[=\rho \left[ \frac{V}{{{v}_{2}}} \right]{{({{v}_{1}}+{{v}_{2}})}^{2}}\]
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