A 0.5kg block moving at a speed of 12ms1 compresses a spring through a distance 30cm when its speed is halved. The spring constant of the spring will be…………Nm1.

by conservation of mechanical energy

              K.E_i + P.E_i = K.E_f + P.E_f

              $\frac{1}{2}\times 0.5v^2+0=\frac{1}{2}\times 0.5{\left(\frac{v}{2}\right)}^2+\frac{1}{2}k{x}^2$  

              $\Rightarrow \frac{1}{2}\times 0.5\left[v^2-\frac{v^2}{4}\right]=\frac{1}{2}k{x}^2$  

              $\Rightarrow \frac{1.5}{4}12\times 12=\frac{k\times 9}{100}$  

              $\frac{1.5\times 36}{9}\times 100=k$  

              k = 600 N/m¹

<p>by conservation of mechanical energy</p><p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; K.E_i + P.E_i = K.E_f + P.E_f</p><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1757396685phpy2wDhw.jpeg" alt="" width="137" height="84" loading="lazy"></picture></div></div><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1757396719phpFTrW0i.jpeg" alt="" width="169" height="70" loading="lazy"></picture></div></div><div><div><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1757396745phpmQO5Mj.jpeg" alt="" width="197" height="109" loading="lazy"></picture></div></div><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mi>k</mi> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math> </span>&nbsp;</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span contenteditable="false"> <math> <mrow> <mo>⇒</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> <mrow> <mo> [</mo> <mrow> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mfrac> <mrow> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mi>k</mi> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </math> </span>&nbsp;</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span contenteditable="false"> <math> <mrow> <mo>⇒</mo> <mfrac> <mrow> <mn>1</mn> <mo>.</mo> <mn>5</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mn>1</mn> <mn>2</mn> <mo>×</mo> <mn>1</mn> <mn>2</mn> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mo>×</mo> <mn>9</mn> </mrow> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mrow> </mfrac> </mrow> </math> </span>&nbsp;</p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>.</mo> <mn>5</mn> <mo>×</mo> <mn>3</mn> <mn>6</mn> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mo>×</mo> <mn>1</mn> <mn>0</mn> <mn>0</mn> <mo>=</mo> <mi>k</mi> </mrow> </math> </span>&nbsp;</p><p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; k = 600 N/m¹</p>

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