Question

Short answer

The minimum velocity of the meteorite is \(1434.1\;{\rm{m/s}}\).

Step-by-step solution

Given data

The temperature is \(T = - 105^\circ {\rm{C}}\).

Understanding the velocity of the meteorite  

In this problem, the final velocity of the meteorite will be zero because it completely melts. Also, all initial kinetic energy is consumed in heating the iron and then melting it.

Calculation of the minimum velocity of the meteorite

The relation to calculate the minimum velocity is given by:

\(\begin{aligned}{c}KE = Q\\\frac{1}{2}m{v_{\rm{i}}}^2 = mc\left( {{T_0} - T} \right) + mL\\\frac{1}{2}{v_{\rm{i}}}^2 = c\left( {{T_0} - T} \right) + L\end{aligned}\)

Here,\({T_0}\)is the melting temperature of iron, cis the specific heat of the iron, mis the mass,\({v_{\rm{i}}}\)is the minimum velocity, KEis the kinetic energy, Qis the heat transfer, and Lis the latent heat.

On plugging the values in the above relation, you get:

\(\begin{aligned}{c}\frac{1}{2}{v_{\rm{i}}}^2 = \left( {450\;{\rm{J/kg}} \cdot ^\circ {\rm{C}}} \right)\left( {1538^\circ {\rm{C}} - \left( { - 105^\circ {\rm{C}}} \right)} \right) + \left( {2.89 \times {{10}^5}\;{\rm{J/kg}}} \right)\\{v_{\rm{i}}} = 1434.1\;{\rm{m/s}}\end{aligned}\)

Thus, \({v_{\rm{i}}} = 1434.1\;{\rm{m/s}}\) is the required minimum velocity.