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Question

A long block A is at rest on a smooth horizontal surface. A small block B, whose mass is half of A, is placed on A at one end and projected along A with some velocity u. The coefficient of friction between the block is μ.


  1. The blocks will reach a final common velocity u3
  2. The work done against friction is two - thirds of the initial kinetic energy of B. 
  3. Before the blocks reach a common velocity, the acceleration of A relative to B is 23μg
  4. Before the blocks reach a common velocity the acceleration of A relative to B is 32μg


Solution

The correct options are
A The blocks will reach a final common velocity u3
B The work done against friction is two - thirds of the initial kinetic energy of B. 
D Before the blocks reach a common velocity the acceleration of A relative to B is 32μg
Let v be the final common velocity,
Using conservation of momentum,
mu=(m+2m)v,
v=u3
Work done by friction is change in loss of kinetic energy, so
12mu212(3m)u29=12mu2[113]=23×12mu2.
The work done agianst friction is two - thirds of the initial kinetic energy of B

The force of friction between the blocks is μmg. 
Acceleration of A(to the right) =a1=μmg2m=μg2
Acceleration of B (to the left)=a2=μmgm=μg
Acceleration of A relative to B=a1(a2)=32μg.

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