(a)
The total kinetic energy of the two blocks.
(a)
Answer to Problem 42P
The total kinetic energy of the two blocks is
Explanation of Solution
Given:
The mass of block 1 is
The velocity of the block 1 in positive x direction is
The mass of block 2 is
The velocity of block 2 in positive x direction is
Formula used:
The expression for kinetic energy is given by,
Here,
Calculation:
The total Kinetic energy of the two blocks can be calculated as,
Further solving the above equation,
Conclusion:
Therefore, the total kinetic energy of the two blocks is
(b)
The velocity of centre of mass of the two block system.
(b)
Answer to Problem 42P
The velocity of centre of mass of the two block system is
Explanation of Solution
Given:
The mass of block 1 is
The velocity of the block 1 in positive x direction is
The mass of block 2 is
The velocity of block 2 in positive x direction is
Formula used:
The expression for momentum is given by,
Calculation:
Since, the momentum of the system remains conserved and can be calculated as,
Conclusion:
Therefore, the velocity of centre of mass of the two block system is
(c)
The velocity of each block relative to the centre of mass.
(c)
Answer to Problem 42P
The velocity of the block 1 relative to the centre of mass is
Explanation of Solution
Given:
The velocity of the block 1 in positive x direction is
The velocity of block 2 in positive x direction
Formula used:
The expression for the velocity with respect to centre of mass is given as,
Here,
Calculation:
The velocity of block 1 with respect to the centre of mass can be calculated as,
The velocity of block 2 with respect to the centre of mass can be calculated as,
Conclusion:
Therefore, The velocity of the block 1 relative to the centre of mass is
(d)
The kinetic energy of the blocks relative to the centre of mass.
(d)
Answer to Problem 42P
The kinetic energy of the blocks relative to the centre of mass is
Explanation of Solution
Given:
The velocity of block 1 in positive x direction is
The velocity of block 2 in positive x direction is
Formula used:
The expression for kinetic energy is given by,
Calculation:
The total Kinetic energy of the two blocks can be calculated as,
Further solving the above equation,
Conclusion:
Therefore, the kinetic energy of the blocks relative to the centre of mass is
(e)
The proof that the kinetic energy in part (a) is greater than the kinetic energy in part (d) by an amount equal to the kinetic energy with respect to centre of mass.
(e)
Answer to Problem 42P
The kinetic energy in part (a) is greater than the kinetic energy in part (d) by an amount equal to the kinetic energy with respect to centre of mass.
Explanation of Solution
Given:
The mass of block 1 is
The velocity of the block 1 in positive x direction is
The mass of block 2 is
The velocity of block 2 in negative x direction
Formula used:
The expression for kinetic energy is given by,
Calculation:
The total Kinetic energy of the two blocks can be calculated as,
The above result is equal to
Conclusion:
Therefore, the kinetic energy in part (a) is greater than the kinetic energy in part (d) by an amount equal to the kinetic energy with respect to centre of mass.
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