Question
A light spring of length $20\, cm$ and force constant $2\, kg/cm$ is placed vertically on a table. A small block of mass $1\, kg$. falls on it. The length $h$ from the surface of the table at which the ball will have the maximum velocity is ............... $\mathrm{cm}$

Answer

Let the spring be compressed by a distance of "x" at a certain moment(as shown in the figure attached below).

Therefore, by using the law of conservation of energy at this moment, we can write

$\operatorname{mg}(h+x)=1 / 2 m v^{2}+1 / 2 k x^{2}$

$\Rightarrow 1 / 2 mv ^{2}= mg ( h + x )-1 / 2 kx ^{2} \ldots$ (i)

[The velocity of the block will be maximum when the kinetic energy is maximum. Also, the velocity keeps on increasing as long its acceleration is increasing]

$\therefore d \left[1 / 2 mv ^{2}\right] / d x = mg - k x =0$

$\Rightarrow mg - kx =0$

$\Rightarrow x=m g / k$

Now, to get the height "h" from the surface of the table at which the velocity is maximum, we need to subtract the distance of compression in the spring from the length of the spring, so,

$h = I - mg / k \ldots \ldots .$ [from (i) $]$

$\Rightarrow h=20-\left[1^{*} 9.8 / 2\right] \ldots \ldots\left[\operatorname{taking} g=9.8 m / s ^{2}\right]$

$\Rightarrow h=20-4.9$

$\Rightarrow h =15.1 cm =15 cm$

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