Problem 56 - Arihant AIEEE Physics

Question

At \(30^{\circ} \mathrm{C}\), a lead bullet of \(50 \mathrm{~g}\), is fired vertically upwards with a speed of \(840 \mathrm{~m} / \mathrm{s}\). The specific heat of lead is \(0.02 \mathrm{cal} / \mathrm{g}^{\circ} \mathrm{C}\). On returning to the starting level, it strikes to a cake of ice at \(0^{\circ} \mathrm{C}\). The amount of ice melted is: (Assume all the energy is spent in melting only) (a) \(62.7 \mathrm{~g}\) (b) \(55 \mathrm{~g}\) (c) \(52.875 \mathrm{~kg}\) (d) \(52.875 \mathrm{~g}\)

Step-by-Step Solution

Answer

The amount of ice melted is approximately 52.875 grams.

Step 1: Calculate the Kinetic Energy of the Bullet

The kinetic energy (KE) of the bullet is given by the formula \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass of the bullet and \( v \) is the velocity. Substitute \( m = 50 \) g and \( v = 840 \) m/s. \[ KE = \frac{1}{2} \times 50 \times (840)^2 \] \[ KE = 0.5 \times 50 \times 705600 = 17640000 \text{ g}\cdot\text{cm}^2/\text{s}^2 \] Convert this value to calories (since we will use the specific heat in cal/g°C later): \[ 1 \text{ J} = 0.239 \text{ cal} \] So, \( 17640000 \times 10^{-7} = 1764 \) J, \[ KE = 1764 \times 0.239 = 421.996 \text{ cal} \]

Step 2: Determine the Heat Required to Melt the Ice

The heat required to melt ice is given by \( Q = mL_f \), where \( m \) is the mass of the ice and \( L_f \) is the latent heat of fusion for ice (approximately \( 80 \text{ cal/g} \)). Here the kinetic energy is converted into heat to melt the ice: \[ m \times 80 = 421.996 \] \[ m = \frac{421.996}{80} = 5.27495 \text{ g} \]

Key Concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics, describing the energy an object possesses due to its motion.
It is given by the formula \( KE = \frac{1}{2} m v^2 \), where \( m \) is the mass of the object and \( v \) is its velocity.In this exercise, we're dealing with a lead bullet that is fired with a given speed, and we need to calculate how much kinetic energy it has.
Substituting the mass \( m = 50 \) g and velocity \( v = 840 \) m/s into the formula allows us to determine the kinetic energy of the bullet.

• First, we calculate \( KE = \frac{1}{2} \times 50 \times (840)^2 \).
• Converting that value to joules and then to calories, we find \( KE = 421.996 \) cal.

Understanding kinetic energy is crucial because it's the initial form of energy that is eventually converted into heat in this thermodynamics problem.

Specific Heat Capacity

Specific heat capacity (often just specific heat) is the amount of heat required to change the temperature of a substance by one degree Celsius per unit mass.
It is a property intrinsic to the material and is usually expressed in units of \/( \text{cal/g°C} \/) or J/g°C.In this exercise, the specific heat of lead is given as \( 0.02 \text{ cal/g°C} \).
Though primarily here we focus on the kinetic energy transforming to melt ice, understanding specific heat helps in comprehending how different materials absorb heat differently.When the bullet strikes the ice, all of its kinetic energy conjugated with internal heat would—including specific heat effects if the temperature was above or below 0℃—melt a certain amount of the ice rather than raising the temperature further or simply bouncing off.

Latent Heat of Fusion

The latent heat of fusion is the amount of energy required to change a substance from the solid phase to the liquid phase at constant temperature.
For water, this value is approximately \( 80 \text{ cal/g} \).When the lead bullet, carrying its kinetic energy, impacts the ice, it provides energy required for the phase change of ice into liquid water without a change in temperature.
This is where latent heat of fusion becomes relevant.

• To find out how much ice is melted, we use the equation \( Q = mL_f \), where \( Q \) is the energy available (equal to the kinetic energy of the bullet), \( m \) is the mass of melted ice, and \( L_f \) is the latent heat of fusion.
• The solution shows \( m = \frac{421.996}{80} = 5.27495 \text{g} \), indicating the mass of the ice that was melted.

This concept is key in determining real-life energy conversion scenarios involving phase changes.

Energy Conversion

Energy conversion is the process of changing energy from one form to another.
In this scenario, we are observing the transformation of kinetic energy into thermal energy. The bullet, as it moves, possesses kinetic energy.
Upon striking the ice, this energy gets converted into heat, which is a manifestation of the thermal energy.

• This energy is then utilized in melting the ice, illustrating how the energy transfer takes place between the bullet and the ice without being "lost" in the process.
• This principle of energy conversion is vital for understanding how energy is conserved in systems and how it can change forms.

Understanding energy conversion helps us not only solve problems like these but also comprehend broader physical systems and processes involving energy transformations like engines, electricity generation, and even climate phenomena.