A lead bullet is traveling at 200 m/s strikes a tree and comes to stop. If half of the heat produced is retained by the bullet, by how much does the temperature increase?
heat energy gained = kinetic energy
m x c x t = K.E.
m x c x t = 1/2 x m x v x v
c x t = 1/2 x v x v
t = (1/2 x v x v)/c
t = (1/2 x 200 x 200)/150 (since, shc = 150 J/kgC)
t = 20000/150
t = 133.33 c
Well, let's see. When a lead bullet strikes a tree and stops, it definitely feels the heat! As for the temperature increase, we'll have to put on our thinking caps... or should I say, clown wig! 🤡
Now, since half of the heat produced is retained by the bullet, we can assume that the other half is dissipated into the surroundings. So, the retained heat is our main focus.
To determine the temperature increase, we need to know the specific heat capacity of lead. However, since I'm more of a joker than a chemist, I'll have to rely on a punchline instead.
Why did the lead bullet go see a doctor after hitting the tree? Because it felt feverish from the heat!
So, while I can't calculate the exact temperature increase for you, I hope my clownish humor did bring a smile to your face! 🤡😄
To calculate the temperature increase, you can use the equation for heat:
heat = mass × specific heat capacity × change in temperature
Given that half of the heat produced is retained by the bullet, we can express the equation as:
0.5 × heat = mass × specific heat capacity × change in temperature
Since the bullet comes to a stop, we know that its initial kinetic energy is converted into heat energy. The initial kinetic energy can be calculated using the formula:
kinetic energy = 0.5 × mass × (velocity)^2
Since the bullet comes to a stop, its final velocity is 0 m/s, so the equation becomes:
0.5 × mass × (200 m/s)^2 = heat
We can simplify this equation to:
mass × 40,000 = heat
Substituting this into the first equation:
0.5 × (mass × 40,000) = mass × specific heat capacity × change in temperature
20,000 = specific heat capacity × change in temperature
Finally, rearranging the equation to solve for the change in temperature:
change in temperature = 20,000 / specific heat capacity
So, the change in temperature depends on the specific heat capacity of the bullet material.
To calculate the increase in temperature, we can make use of the conservation of mechanical energy principle. The kinetic energy of the bullet is converted entirely into heat, which can be expressed as:
ΔE = Q = mcΔT
Where:
ΔE = change in mechanical energy (kinetic energy of the bullet)
Q = heat produced
m = mass of the bullet
c = specific heat capacity of the bullet material
ΔT = change in temperature
We are given that half of the heat produced is retained by the bullet. Therefore, the heat produced will be:
Q = 0.5 * mcΔT
Since the bullet comes to a stop, all of its kinetic energy is converted into heat. The kinetic energy can be calculated using the formula:
K.E. = (1/2)mv^2
Where:
K.E. = kinetic energy
m = mass of the bullet
v = velocity of the bullet
Substituting this expression into the equation for heat produced, we get:
(1/2)mv^2 = 0.5 * mcΔT
Canceling out the mass from both sides, we can simplify the equation to:
(1/2)v^2 = 0.5cΔT
Now, we can solve for ΔT:
ΔT = (1/0.5)(v^2/c)
Substituting the given values, we have:
ΔT = (1/0.5)(200^2/0.5)
Performing the calculations, we find:
ΔT = 200^2*2
ΔT = 80,000 K
Therefore, the temperature of the bullet increases by 80,000 Kelvin.