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Physics-

General

Easy

Question

The blocks $A$ and $B$ , each of mass $m$ , are connected by massless spring of natural length $L$ and spring constant $k$ . The blocks are initially resting on a smooth horizontal floor with the spring at its natural length, as shown in figure. A third identical block $C$ , also of mass $m$ , moves on the floor with a speed $v$ along the line joining $A$ and $B$ , and collides with $A$ . Then

The KE o the $A - B$ system, at maximum compression of the spring, is zero
The KE of the $A - B$ system, at maximum compression of the spring is $\frac{1}{4} m v^{2}$
The maximum compression of the spring is $v \sqrt{\frac{m}{k}}$
The maximum compression of the spring is $v \sqrt{\frac{m}{2 k}}$

The correct answer is: The maximum compression of the spring is $v square root of fraction numerator m over denominator 2 k end fraction end root$

The compression of spring is maximum when velocities of both blocks $A$ and $B$ is same. Let it be $v subscript 0 end subscript$ , then from conservation law of momentum
$m v equals m v subscript 0 end subscript plus m v subscript 0 end subscript equals 2 m v subscript 0 end subscript rightwards double arrow v subscript 0 end subscript equals fraction numerator v over denominator 2 end fraction$
$therefore$ kinetic energy of $A minus B$ system at that stage
$equals fraction numerator 1 over denominator 2 end fraction open parentheses m plus m close parentheses cross times open parentheses fraction numerator v over denominator 2 end fraction close parentheses to the power of 2 end exponent equals fraction numerator m v to the power of 2 end exponent over denominator 4 end fraction$
Further loss in KE> = gain in elastic potential energy
$i e$ , $fraction numerator 1 over denominator 2 end fraction m v to the power of 2 end exponent minus fraction numerator 1 over denominator 4 end fraction m v to the power of 2 end exponent equals fraction numerator 1 over denominator 4 end fraction m v to the power of 2 end exponent equals fraction numerator 1 over denominator 2 end fraction k x to the power of 2 end exponent$
$rightwards double arrow blank x equals v square root of fraction numerator m over denominator 2 k end fraction end root$

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