A body of mass 10 kg explodes into three masses in the ratio 2 : 3 : 5. Two light fragment fly off at right angle with velocity 12m/ s and 8m/s. Find the velocity of third fragment.

After explosion,

Let 2 kg mass after explosion fly off along X-axis and 3 kg mass fly off along Y-axis perpendicular to direction of motion of 2 kg mass.

Suppose the velocity of third fragment after explosion is 

∴ Momentum of 2 kg mass is, 

              

Momentum of 3 kg mass is, 

             

Momentum of 5 kg mass is, 

           

According to the law of conservation of linear momentum, 

               

          

          

∴                                 

                     

This is the required velocity of the third fragment. 

Let us consider, rocket of mass m₀ take off with velocity v₀ from ground. Let the fuel burnt at the rate of dm/dt and burnt gases eject with velocity u w.r.t. rocket.
Let at any instant t, v be the velocity of rocket and m be the mass of rocket. Therefore velocity of burnt gas ejected w.r.t. ground is (v – u). As the fuel burns at the rate of dm/dt, therefore in time dt, dm mass of the fuel will burn and velocity of rocket increases by dv.

Here m is the mass of rocket and dm is the mass of gas ejected. As the fuel burns the mass of rocket decreases. Therefore, to calculate the velocity of rocket at any instant in terms of mass of rocket, dm must be replaced by ‘dm’.

Let the cart be connected to horse through a string. Let  be the tension in the string. To start the motion, the horse presses the ground slantingly and the horse gets equal and opposite reaction 


Let  be the force of friction between cart and ground and let the system accelerate with acceleration 
      Resolve  into two rectangular components 
                 From the free body diagram of horse,
                                     V = mg
and                       H - T = ma                       ...(1)
       From the free body diagram of the cart,
                           T - F = Ma                          ...(2)
Adding (1) and (2), we have,
                          H - F = (m+M)a
or                       
    The system will move if and only if a>0
i.e.                      H>F