The following questions which appeared in All India Pre-Medical/Pre-Dental Entrance Examination (AIPMT) 2008 are worth noting:

**(i)** A shell of mass 200 g is ejected from a gun of mass 4 kg by an explosion that generates 1.05 kJ of energy. The initial velocity of the shell is

(1) 80 ms^{–1}

(2) 40 ms^{–1}

(3) 120 ms^{–1}

(4) 100 ms^{–1}

This question is meant for testing your grasp of the laws of conservation of momentum and energy. If *v* and *V* represent the initial velocity of the shell of mass *m *and the recoil velocity of the gun of mass *M* respectively, we have (from the law of conservation of momentum)

*mv* + *MV* = 0 from which *V* = – (*m/M*)*v* = – (0.2/4)*v* = – *v/*20 ms^{–1}

[The negative sign shows that the recoil velocity of the gun is opposite to the velocity of the shell].

The energy of the explosion is used to impart kinetic energy to the shell and the gun so that we have

½ (*mv*^{2} + *MV*^{2}) = 1.05×10^{3}^{}

Or, ½ [0.2 *v*^{2} + (4*v*^{2}/400^{ })] = 1.05×10^{3}^{}

This gives *v*^{2} = 10^{4} so that *v* = 100 ms^{–1}.

**(ii) **Sand is being dropped on a conveyor belt at the rate of *M* kg/s. The force necessary to keep the belt moving with a constant velocity of *v* ms^{–1} will be

(1) 2*Mv* newton

(2) *Mv/*2 newton

(3) zero* *

(4) *Mv *newton

Since there is a time rate of change of momentum equal to *Mv* per second, the force required is *Mv *Newton.

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