A metal has a latent heat of fusion of $2.32 \times 10 ^ { 4 } \mathrm {~J} / \mathrm { kg } _ { 1 }$ a specific heat of $128 \mathrm {~J} / \mathrm { kg } \cdot \mathrm { K } _ { \mathrm { r } }$ and a melting point of $228 ^ { \circ } \mathrm { C } \text {. }$ A 30-g pellet of this metal at $16 ^ { \circ } \mathrm { C }$ hits a solid wall and comes to a complete stop. What would the speed of the pellet have to be in order for it to melt completely when it hits the wall, assuming that all of its kinetic energy is transformed into heat within the pellet?

Question

A metal has a latent heat of fusion of $2.32 \times 10 ^ { 4 } \mathrm {~J} / \mathrm { kg } _ { 1 }$ a specific heat of $128 \mathrm {~J} / \mathrm { kg } \cdot \mathrm { K } _ { \mathrm { r } }$ and a melting point of $228 ^ { \circ } \mathrm { C } \text {. }$ A  30-g pellet of this metal at $16 ^ { \circ } \mathrm { C }$ hits a solid wall and comes to a complete stop. What would the speed of the pellet have to be in order for it to melt completely when it hits the wall, assuming that all of its kinetic energy is transformed into heat within the pellet?

Quizplus · Accepted Answer

To find the speed necessary for the pellet to melt completely, we first calculate the total heat required to raise the temperature of the pellet to its melting point and then to melt it completely. The heat required to raise the temperature, $Q_1$, is given by $Q_1 = mc\Delta T$, where $m = 0.03 \, \text{kg}$ (since 30 g = 0.03 kg), $c = 128 \, \text{J/kg}\cdot\text{K}$, and $\Delta T = 228^\circ \text{C} - 16^\circ \text{C} = 212 \, \text{K}$. The heat required to melt the pellet, $Q_2$, is given by $Q_2 = mL$, where $L = 2.32 \times 10^4 \, \text{J/kg}$. The total heat required is $Q_{\text{total}} = Q_1 + Q_2$. This heat comes from the kinetic energy of the pellet, which is $\frac{1}{2}mv^2$. Solving for $v$ gives the speed necessary for the pellet to melt completely.

A Metal Has a Latent Heat of Fusion Of 2.32×104 J/kg12.32 \times 10 ^ { 4 } \mathrm {~J} / \mathrm { kg } _ { 1 }2.32×104 J/kg1​

A Metal Has a Latent Heat of Fusion Of $2.32 \times 10 ^ { 4 } \mathrm {~J} / \mathrm { kg } _ { 1 }$