A lead bullet penetrates into a solid object and melts. Assuming that 40% of its kinetic energy is used to heat it, the initial speed of bullet is:

(Given, initial temperature of the bullet = 127°C,

Melting point of the bullet = 327°C,

Latent heat of fusion of lead = 2.5 × 10⁴ J kg^-¹,

Specific heat capacity of lead = 125 J/kg K)

Option 1 -

125 ms^-1

Option 2 -

500 ms^-1

Option 3 -

250 ms^-1

Option 4 -

600 ms^-1

2 Views | Posted 3 months ago

Asked by Shiksha User

1 Answer

Answered by

Payal Gupta | Contributor-Level 10

3 months ago

Correct Option - 2

Detailed Solution:

40% of K.E. = $m c Δ T + m L$

$0.4 \times \frac{1}{2} \times m v^{2} = m [125 (327 - 127) + 2.5 \times 1 0^{4}]$

$\Rightarrow 0.2 v^{2} = [250 \times 1 0^{2} + 25 \times 1 0^{3}]$

$v^{2} = \frac{2 \times 25 \times 1 0^{3}}{0.2}$

v = 5 × 100

v = 500 m/sec

40% of K.E. = <math> <mrow> <mi>m</mi> <mi>c</mi> <mi>Δ</mi> <mi>T</mi> <mo>+</mo> <mi>m</mi> <mi>L</mi> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1813496944"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mn>0</mn> <mo>.</mo> <mn>4</mn> <mo>×</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mi>m</mi> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>m</mi> <mrow> <mo> [</mo> <mrow> <mn>1</mn> <mn>2</mn> <mn>5</mn> <mrow> <mo> (</mo> <mrow> <mn>3</mn> <mn>2</mn> <mn>7</mn> <mo>−</mo> <mn>1</mn> <mn>2</mn> <mn>7</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mo>.</mo> <mn>5</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1813496945"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mo>⇒</mo> <mn>0</mn> <mo>.</mo> <mn>2</mn> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo> [</mo> <mrow> <mn>2</mn> <mn>5</mn> <mn>0</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mn>5</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1813496946"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <msup> <mrow> <mi>v</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mn>5</mn> <mo>×</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>0</mn> <mo>.</mo> <mn>2</mn> </mrow> </mfrac> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1813496947"> </o:OLEObject></xml><! [endif]-> v = 5 × 100v = 500 m/sec

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