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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 the 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⁻¹ K⁻¹)

Asked in JEE Main 27th June 2nd Shift 2022 · Mechanical work converted into heat

• (1) 125 m s⁻¹
• (2) 500 m s⁻¹
• (3) 250 m s⁻¹
• (4) 600 m s⁻¹

Step-by-step solution

Why the other options are wrong

• (1) comes from no consistent slip lands here; it is a filler a quarter of the key. The routes worth checking all miss it: dropping either the latent or the sensible term gives 354 m s⁻¹, and dropping the 40% gives 316 m s⁻¹.
• (3) comes from putting the 1/2 the wrong way up, writing 0.4×2mv² instead of 0.4×1/2mv². That gives 0.8v²=5×10⁴ and v=250 m s⁻¹, exactly half the key.
• (4) comes from another filler above the key. The heat needed per kilogram is fixed at 5×10⁴ J, and 600 m s⁻¹ would supply 0.2×(600)²=7.2×10⁴ J kg⁻¹ — more than the bullet can use.

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