A 30.0-g lead bullet leaves a rifle at a temperature of 89.0° C and hits a steel plate. If the bullet melts, what is the minimum speed it must have?

Latent heat of melting r = 22400 J/kg, specific heat c =130 J/kg•oC

Q = m•c•Δt + r•m = m•(c•Δt + r) =
= 0.03•( 130•(327-89) + 22400) =1600 J.
m•v^2/2 = Q
v = sqrt (2•Q/m) =sqrt (2• 1600/0.03) =326 m/s.

327

To determine the minimum speed at which the lead bullet must leave the rifle in order to melt upon impact with the steel plate, we need to consider the energy transferred to the bullet.

First, let's recall the concept of heat transfer. The heat transferred (Q) can be calculated using the heat equation:

Q = mcΔT

Where:
- Q is the heat transferred
- m is the mass of the lead bullet
- c is the specific heat capacity of lead
- ΔT is the change in temperature (final temperature - initial temperature)

In this case, since the bullet melts, the heat transferred should be equal to the heat of fusion (ΔHf) of lead. The heat of fusion is the amount of energy required to change a substance from a solid to a liquid state.

Now, we need to calculate the heat transferred to melt the bullet.

First, let's determine the heat capacity of the lead bullet. The specific heat capacity (c) of lead is approximately 0.13 J/g°C.

Q = mcΔT
Q = (30.0 g)(0.13 J/g°C)(Tf - Ti)

Where:
- Tf is the final temperature (at which the lead bullet melts)
- Ti is the initial temperature (89.0°C)

Since the bullet melts, its final temperature will be its melting point, which is approximately 327.5°C for lead.

Q = (30.0 g)(0.13 J/g°C)(327.5°C - 89.0°C)

Now, we know that the heat transferred (Q) is equal to the heat of fusion (ΔHf) of lead:

Q = ΔHf

ΔHf = (30.0 g)(0.13 J/g°C)(327.5°C - 89.0°C)

Next, we need to calculate the kinetic energy (KE) of the lead bullet. The kinetic energy formula is:

KE = 0.5mv^2

Where:
- KE is the kinetic energy
- m is the mass of the bullet (30.0 g)
- v is the velocity (speed) of the bullet

To find the minimum speed at which the bullet must leave the rifle, we want to determine the velocity (v) needed to provide the same amount of energy as the heat of fusion (ΔHf):

KE = ΔHf
0.5mv^2 = (30.0 g)(0.13 J/g°C)(327.5°C - 89.0°C)

Simplifying the equation:

0.5v^2 = (30.0 g)(0.13 J/g°C)(327.5°C - 89.0°C)

Now, we can solve the equation for v by isolating it:

v^2 = (2[(30.0 g)(0.13 J/g°C)(327.5°C - 89.0°C)]) / 0.5

Finally, take the square root of both sides to find the minimum speed:

v = √[(2[(30.0 g)(0.13 J/g°C)(327.5°C - 89.0°C)]) / 0.5]

Evaluating this equation will give you the minimum speed in the appropriate unit, likely meters per second (m/s).