question_answer

An explosion blows a rock into three pieces. Two pieces whose masses are 200 kg and 100 kg go off at right angles to each other with a velocity of 8 m/s and 12 m/s respectively. If the third piece flies off with a velocity of 25 m/s, then calculate the mass of this piece and indicate the direction of flight of this piece in a diagram. 

Answer:

                  Suppose the explosion of the rock occurs at O. One piece of mass 100 kg goes along go off, other piece of mass 200 kg goes along \[OB\bot OA\]. Let the third piece of mass m fly off at a speed of \[25\,m/s\]along OC, at \[\angle \theta \]with OA, Fig. 3 (HT). 14.    Applying principle of conservation of linear momentum along OA \[100\times 12-m\upsilon \cos \theta =0\] \[1200=m\upsilon \cos \theta =25m\,\cos \theta \] \[\therefore \]  \[m\cos \theta =\frac{1200}{25}=48\]                                                     .... (i) Again, applying principle of conservation of linear momentum along OB \[200\times 8-m\upsilon \sin \theta =0\] \[1600=m\upsilon \sin \theta =25\,m\,\sin \theta \] \[\therefore \] \[m\,\sin \theta =\frac{1600}{25}=64\]                                                                   .... (ii) Square (i) and (ii),attd add \[{{m}^{2}}={{48}^{2}}+{{64}^{2}}\] m = 80 kg Divide (ii) by (i), \[\tan \theta =\frac{64}{48}=\frac{4}{3}\] \[\theta ={{\tan }^{-1}}\left( 4/3 \right)={{53}^{o}}\]