A 15 g lead bullet is tested by firing it into a fixed block of wood with a mass of
1.05 kg. The block and the embedded bullet together absorb all the heat generated. After
thermal equilibrium has been reached, the system has a change in temperature
measured as 0.020 oC. Estimate the entering speed of the bullet.
(Hint: Use the fact that energy is conserved)
The bullet/block collision, the bullet initial energy is 1/2 m v^2
The energy converts to heat, mc*deltatemp
So the question is, what is the specific heat of the lead/wood combination. Well, you can learn something here. Do them separately
1/2 massbullet*v^2=masswood*cwood*deltaT + masslead*clead*deltaT
http://www.engineeringtoolbox.com/specific-heat-solids-d_154.html
http://www.sciencebyjones.com/specific_heat1.htm
solve for velocity. What is interesting, is where most of the energy went: the lead, or the wood.
To solve this problem, we can use the principle of conservation of energy. The initial kinetic energy of the bullet is equal to the final thermal energy of the bullet and the block.
1. Start by determining the initial kinetic energy of the bullet. The formula for kinetic energy is:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of the bullet, and v is the velocity of the bullet.
2. Next, calculate the final thermal energy of the bullet and the block. The thermal energy is given by:
Q = m_total * c * ΔT
where Q is the thermal energy, m_total is the total mass of the bullet and the block, c is the specific heat capacity, and ΔT is the change in temperature.
For this problem, the bullet and the block are in thermal equilibrium, which means that their final temperature is the same. We can assume that the specific heat capacities are approximately the same, so we can simplify the equation to:
Q = (m_bullet + m_block) * c * ΔT
3. Since the initial kinetic energy is equal to the final thermal energy, we can set KE equal to Q:
(1/2) * m_bullet * v^2 = (m_bullet + m_block) * c * ΔT
4. Rearrange the equation to solve for v, the velocity of the bullet:
v = sqrt((2 * (m_bullet + m_block) * c * ΔT) / m_bullet)
5. Substitute the given values into the equation:
m_bullet = 0.015 kg
m_block = 1.05 kg
c = specific heat capacity (unknown)
ΔT = 0.020 oC
v = sqrt((2 * (0.015 kg + 1.05 kg) * c * 0.020 oC) / 0.015 kg)
6. Now, we need to know the specific heat capacity of lead. The specific heat capacity of lead is approximately 130 J/(kg·°C).
v = sqrt((2 * (0.015 kg + 1.05 kg) * 130 J/(kg·°C) * 0.020 oC) / 0.015 kg)
7. Calculate the value of v.
v = sqrt((2 * (1.065 kg) * 130 J/(kg·°C) * 0.020 oC) / 0.015 kg)
v = sqrt(2.172 kg·J/°C)
The entering speed of the bullet is approximately 1.474 m/s.
Therefore, the estimated entering speed of the bullet is approximately 1.474 m/s.
To estimate the entering speed of the bullet, we can use the principle of conservation of energy.
First, let's identify the different forms of energy in this system:
1. Kinetic energy (KE) of the bullet before impact.
2. Heat energy absorbed by the block and bullet.
3. Internal energy change due to the change in temperature.
The total energy before and after the collision remains the same. Therefore, the sum of the bullet's kinetic energy and the heat energy absorbed by the block and bullet will equal the internal energy change due to the temperature change.
The kinetic energy of the bullet can be expressed as:
KE = (1/2) * m * v^2
where m is the mass of the bullet and v is the unknown velocity.
The heat energy absorbed by the block and bullet can be calculated using the specific heat capacity (c) of wood and the mass (M) of the block and bullet:
Heat absorbed = c * M * ΔT
where ΔT is the change in temperature given as 0.020 oC.
Setting up the conservation of energy equation:
KE + Heat absorbed = Internal energy change
We can plug in the known values:
(1/2) * m * v^2 + c * M * ΔT = 0
Rearranging the equation to solve for v:
v = √((-2 * c * M * ΔT) / m)
Now, we can plug in the values given:
m = 15 g = 0.015 kg
M = 1.05 kg
c = Specific heat capacity of wood (you may need to look up this value)
ΔT = 0.020 oC
By substituting these values into the equation, you can then calculate the entering speed of the bullet.
Note: It's important to ensure that all units are consistent (e.g., mass in kilograms and temperature in Kelvin) to get accurate results.