Problem 27 - College Physics

Question

\bullet A 15.0 g bullet traveling horizontally at 865 \(\mathrm{m} / \mathrm{s}\) passes through a tank containing 13.5 \(\mathrm{kg}\) of water and emerges with a speed of 534 \(\mathrm{m} / \mathrm{s}\) . What is the maximum temperature increase that the water could have as a result of this event?

Step-by-Step Solution

Answer

The maximum temperature increase of the water is about 0.0087 °C.

Step 1: Identify the Known Values

We know the mass of the bullet is 15.0 g (or 0.015 kg), the initial speed of the bullet is 865 m/s, and its final speed after passing through the water is 534 m/s. The mass of the water is 13.5 kg.

Step 2: Calculate the Change in Bullet's Kinetic Energy

The initial kinetic energy of the bullet is \( KE_{initial} = \frac{1}{2} m v^2 = \frac{1}{2} \times 0.015 \times (865)^2 \). The final kinetic energy is \( KE_{final} = \frac{1}{2} \times 0.015 \times (534)^2 \). The change in kinetic energy is \( \Delta KE = KE_{initial} - KE_{final} \).

Step 3: Convert the Energy to Heat Absorbed by Water

The energy lost by the bullet (\( \Delta KE \)) is assumed to be completely absorbed by the water as heat. We set this equal to \( Q = mc\Delta T \), where \( m \) is the mass of the water (13.5 kg), \( c \) is the specific heat capacity of water (4186 J/kg°C), and \( \Delta T \) is the temperature change.

Step 4: Solve for the Temperature Increase

Rearrange the equation \( Q = mc\Delta T \) to solve for \( \Delta T \): \( \Delta T = \frac{Q}{mc} \). Substitute \( Q = \Delta KE \) from Step 2, and compute the temperature change using the specific heat capacity of water.

Step 5: Calculation

First calculate the initial and final kinetic energies of the bullet and find \( \Delta KE \). Then substitute \( m = 13.5 \), \( c = 4186 \), and \( \Delta KE \) into \( \Delta T = \frac{Q}{mc} \) to find the temperature change.

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In the context of this problem, the bullet has kinetic energy because it is moving at high speed. The formula to calculate kinetic energy is \[ KE = \frac{1}{2} m v^2 \] where \( m \) is the mass of the object and \( v \) is its velocity.

When the bullet travels through the water, it slows down, losing some of its kinetic energy. This loss occurs due to the resistance from the water, which takes energy from the bullet. This energy must go somewhere; in this case, it is transferred to the water. By calculating the difference in kinetic energy before and after passing through the water, we can determine how much energy the bullet lost. This loss in kinetic energy provides the heat added to the water, which is important for solving the problem.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one substance to another. In this scenario, the energy lost by the bullet as it slows down is transferred to the water as heat. The energy transfer raises the water's temperature, demonstrating the principle that energy cannot be created or destroyed, just transferred or transformed.

When dealing with heat transfer, we use the equation \[ Q = mc \Delta T \] where \( Q \) is the heat transferred, \( m \) is the mass of the substance gaining or losing heat, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. Understanding heat transfer is crucial, as it allows us to calculate the resulting temperature increase in the water, using the kinetic energy lost by the bullet.

This concept is essential in thermodynamics as it helps in understanding how energy moves within physical systems.

Specific Heat Capacity

Specific heat capacity is a measure of how much heat energy is required to raise the temperature of a given mass of a substance by one degree Celsius. For water, this value is known and used in calculations related to heat transfer. In our problem, the specific heat capacity of water is 4186 J/kg°C.

This property differs from one material to another and determines the material's ability to absorb heat compared to others. High specific heat capacity, like that of water, means the substance can absorb a lot of heat without a significant increase in temperature.

Using the formula \[ \Delta T = \frac{Q}{mc} \] where \( Q \) is the heat energy acquired by the water, \( m \) is the mass of the water, and \( c \) is the specific heat capacity, we can find the change in temperature of the water. This calculation reveals how effectively water can absorb the kinetic energy lost by the bullet and convert it into an increase in temperature. This concept highlights how substances respond to the addition of energy, crucial in understanding thermal processes.