A 0.003 0-kg lead bullet is traveling at a speed of 240 m/s when it embeds in a block of ice at 0°C. If all the heat generated goes into melting ice, what quantity of ice is melted? (Lf = 80 kcal/kg, the specific heat of lead = 0.03 kcal/kg×°C, and 1 kcal = 4 186 J)
first get the KE of the bullet
KE = ½mV² = ½(0.003)(240)² = 86.4J
heat of fusion of ice is 334 kJ/kg
set the KE from above to melting ice
86.4J = 334000 J/kg x M
solve for M
M = 0.0002587 kg of ice (d)
sorry, I use metric (SI) system only, which does not include calories.
A 0.003 0-kg lead bullet is traveling at a speed of 240 m/s when it embeds in a block of ice at 0°C. If all the heat generated goes into melting ice, what quantity of ice is melted?
To find the quantity of ice melted, we need to calculate the heat generated by the bullet and then convert it to the quantity of ice melted.
Step 1: Calculate the heat generated by the bullet.
The heat generated by the bullet is equal to the change in its kinetic energy. The formula to calculate the kinetic energy is:
K.E = (1/2) * mass * velocity^2
Given:
Mass of the bullet (m) = 0.003 kg
Velocity of the bullet (v) = 240 m/s
Substituting the values into the formula:
K.E = (1/2) * 0.003 kg * (240 m/s)^2
K.E = 0.003 kg * 57,600 m^2/s^2
K.E = 172.8 kg⋅m^2/s^2
Step 2: Convert the kinetic energy into heat (in joules).
1 Kcal = 4186 J
So, 1 kg-cal = 4186 J / kg
And, 1 kcal/kg = 4186 J / kg
Given:
1 kcal = 4186 J
Converting the kinetic energy to joules:
Heat generated = 172.8 kg⋅m^2/s^2 * (1 kcal / 4186 J)
Heat generated = 0.0413 kcal * (1 kcal / 4186 J)
Heat generated = 0.0413 * 4186 J
Heat generated = 172.858 J
Step 3: Determine the quantity of ice melted.
To melt ice, the heat generated is equal to the heat required for the phase change (melting), which is:
Q = m * Lf
Given:
Lf (latent heat of fusion of ice) = 80 kcal/kg
Converting Lf from kcal/kg to J/kg:
1 kcal = 4186 J
1 kg-cal = 4186 J / kg
Lf = 80 kcal/kg * (4186 J / 1 kcal)
Lf = 80 * 4186 J/kg
Lf = 334,880 J/kg
Now, we can calculate the quantity of ice melted:
Quantity of ice melted = Heat generated / Lf
Quantity of ice melted = 172.858 J / 334880 J/kg
Quantity of ice melted = 0.000516 kg
Therefore, 0.000516 kg (or approximately 0.52 g) of ice is melted.
To find the quantity of ice that is melted, we need to calculate the heat generated by the bullet and then use it to determine the amount of ice melted. Here's how you can approach this problem:
1. Calculate the kinetic energy (KE) of the bullet:
The formula for kinetic energy is KE = 1/2 * mass * velocity^2.
Convert the bullet's mass from grams to kilograms: 0.0030 kg.
Calculate the kinetic energy of the bullet: KE = 1/2 * 0.0030 kg * (240 m/s)^2.
2. Calculate the heat generated by the bullet:
The heat generated can be found using the formula Q = KE + m * Cp * ΔT.
The bullet transfers its kinetic energy to the ice and raises its temperature from 0°C to the melting point.
The change in temperature (ΔT) is 0°C - 0°C = 0°C.
The specific heat of lead (Cp) is given: 0.03 kcal/kg×°C. Convert it to J/kg×°C: 0.03 kcal/kg×°C * 4186 J/kcal.
Plug these values into the formula: Q = KE + m * Cp * ΔT.
3. Convert the heat generated to kilocalories:
Divide the heat generated (in J) by the conversion factor: 1 J = 0.000239 kcal.
This will give you the heat generated in kilocalories.
4. Calculate the amount of ice melted:
The latent heat of fusion (Lf) for ice is given: 80 kcal/kg.
Divide the heat generated (in kcal) by the latent heat of fusion to find the amount of ice melted.
Step 1: Calculate the kinetic energy (KE) of the bullet:
KE = 1/2 * 0.0030 kg * (240 m/s)^2.
Step 2: Calculate the heat generated by the bullet:
Q = KE + m * Cp * ΔT.
Step 3: Convert the heat to kilocalories:
Q (kcal) = Q (J) * 0.000239 kcal/J.
Step 4: Calculate the amount of ice melted:
Mass of ice melted = Q (kcal) / Lf.
Plug in the given values and follow these steps to find the amount of ice melted.