$1$ kg body explodes into three fragments. The ratio of their masses is $1:1:3$. The fragments of same mass move perpendicular to each other with speeds $30$ m/s, while the heavier part remains in the initial direction. The speed of heavier part is:
Key Idea: Equate the momenta of the system along two perpendicular axes. Let $ u $ be the velocity and $ \theta $ the direction of the third piece as shown.
Equating the momenta of the system along OA and OB to zero, we get $ m\times 30-3\,m\times v\cos \theta =0 $ ... (i) and $ m\times 30-3\,m\times v\sin \theta =0 $... (ii) These give $ 3\,mv\,\cos \theta =3mv\sin \theta $ or $ \cos \theta =\sin \theta $ $ \therefore $$ \theta ={{45}^{o}} $ Thus, $ \angle AOC=\angle BOC={{180}^{o}}-{{45}^{o}}={{135}^{o}} $ Putting the value of $ \theta $ in E (i), we get $ 30\,m=3mv\cos {{45}^{o}}=\frac{3mv}{\sqrt{2}} $ $ \therefore $$ v=10\sqrt{2}\,m/s $ The third piece will move with a velocity of $ 10\sqrt{2}\,m/s $ in a direction making an angle of $ {{135}^{o}} $ with either piece. Alternative: Key Idea: The square of momentum of third piece is equal to sum of squares of momentum of first and second piece. As from key idea, $ {{p}_{3}}^{2}={{p}_{1}}^{2}+{{p}_{2}}^{2} $ or $ {{p}_{3}}=\sqrt{{{p}_{1}}^{2}+{{p}_{2}}^{2}} $ or $ 3m{{v}_{3}}=\sqrt{{{(m\times 30)}^{2}}+{{(m\times 30)}^{2}}} $ or $ {{v}_{3}}=\frac{30\sqrt{2}}{3}=10\sqrt{2}\,m/s $
Work is correlated to force and the displacement over which it acts. When an object is replaced parallel to the force's line of action, it is thought to be doing work. It is a force-driven action that includes movement in the force's direction.
The work done by the force is described to be the product of the elements of the force in the direction of the displacement and the magnitude of this displacement.
Energy:
A body's energy is its potential to do tasks. Anything that has the capability to work is said to have energy. The unit of energy is the same as the unit of work, i.e., the Joule.
There are two types of mechanical energy such as; Kinetic and potential energy.
Power is the rate at which energy is transferred, conveyed, or converted or the rate of doing work. Technologically, it is the amount of work done per unit of time. The SI unit of power is Watt (W) which is joules per second (J/s). Sometimes the power of motor vehicles and other machines is demonstrated in terms of Horsepower (hp), which is roughly equal to 745.7 watts.
Power is a scalar quantity, which gives us a quantity or amount of energy consumed per unit of time but with no manifestation of direction.
