Question

What must the initial speed of a lead bullet be at $\mathbf{25}\mathbf{.}{\mathbf{0}}^{\mathbf{0}}\mathit{C}$so that the heat developed when it is brought to rest will be just sufficient to melt it? Assume that all the initial mechanical energy of the bullet is converted to heat and that no heat flows from the bullet to its surroundings. (Typical rifles have muzzle speeds that exceed the speed of sound in air, which is 347 m/s at $\mathbf{25}\mathbf{.}{\mathbf{0}}^{\mathbf{0}}\mathit{C}$.)

Short answer

The initial speed of the bullet will be 357m/s.

Step-by-step solution

Equation of heat flow

The equation of heat flow for a body of mass m, specific heat capacity c and for a temperature change of $\mathit{\delta }\mathit{T}$is $\mathit{Q}\mathbf{=}\mathit{m}\mathit{c}\mathit{\delta }\mathit{T}$.

The kinetic energy of a body of mass m and moving with velocity v is localid="1665118541610" $\frac{\mathbf{1}}{\mathbf{2}}\mathit{m}{\mathit{v}}^{\mathbf{2}}$.

Equate both the energy equations to find velocity

Given that the whole kinetic energy melts the bullet. So, equate the energies and find the velocity.

$$\begin{gathered} K.E.=Q \\ \frac{1}{2}m{v}^2=mc\delta T+m{L}_f \\ \frac{1}{2}{v}^2=c\delta T+L_f \\ v=\sqrt{2(c\delta T+L_f)} \\ v=\sqrt{\left(2\right[(130\times (327.3-25)+24.5\times {10}^3)} \\ v=357m/s \end{gathered}$$

Hence, the initial speed of the bullet will be 357 m/s.