question_answer

Two ends of a train moving with a constant acceleration passes a certain point with velocities u and v. Show that the velocity with which the middle point of the train passes the same point is\[\sqrt{({{u}^{2}}+{{v}^{2}})/2}\].

Answer:

                Let x be the total length of the train, V be the velocity of the train while passing a certain middle point and a be the uniform acceleration of the train. Taking the motion of the train when middle point is passing from the given point, we have \[\text{u}=\text{u},\text{ v}=\text{V},\text{s}=\text{x}/\text{2 };\text{a}=\text{a}\] Using, \[{{\upsilon }^{\text{2}}}=\text{ }{{\text{u}}^{\text{2}}}+\text{2as}\], we have \[{{\text{V}}^{\text{2}}}={{\text{u}}^{\text{2}}}+\text{2ax}/\text{2}={{\text{u}}^{\text{2}}}+\text{ax}\]                                              ...(i) Taking the motion of train when the last end of train is passing from the given point, then \[\text{u}=\text{u},\text{v}=\text{v},\text{ a}=\text{a},\text{ s}=\text{x}\] Now, we have \[{{\text{v}}^{\text{2}}}={{\text{u}}^{\text{2}}}+\text{2ax}\] or \[\text{ax}=\frac{{{\upsilon }^{2}}-{{u}^{2}}}{2}\] Putting this value in (i), we get \[{{\text{V}}^{\text{2}}}={{\text{u}}^{\text{2}}}+\frac{{{\upsilon }^{2}}-{{u}^{2}}}{2}=\frac{{{u}^{2}}+{{\upsilon }^{2}}}{2}\]or \[\text{V}=\sqrt{({{u}^{2}}+{{\upsilon }^{2}})/2}\]