Question

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

Short answer

Calculate kinetic energy lost by the bullet, assume it's converted to thermal energy, and use it to find the maximum water temperature increase, \( \Delta T \).

Step-by-step solution

Identify the System and Energy Change

We will consider the water and bullet as a single system. The bullet loses kinetic energy when passing through the water, and this energy is transferred to the water, potentially increasing its temperature. The maximum temperature increase will occur if all the lost kinetic energy is transformed into thermal energy of the water.

Calculate Initial and Final Kinetic Energy of the Bullet

The kinetic energy (KE) is calculated using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity.- Initial kinetic energy: \( KE_{i} = \frac{1}{2} \times 0.015 \times (865)^2 \).- Final kinetic energy: \( KE_{f} = \frac{1}{2} \times 0.015 \times (534)^2 \).

Calculate the Change in Kinetic Energy

The change in kinetic energy is the energy lost by the bullet, which is:\[ \Delta KE = KE_{i} - KE_{f} \]

Relate Lost Kinetic Energy to Temperature Increase

The change in kinetic energy \( \Delta KE \) can be used to calculate the temperature increase of the water. The formula \( Q = mc\Delta T \) relates heat added to a substance to its mass \( m \), its specific heat capacity \( c \), and the change in temperature \( \Delta T \).- The specific heat capacity of water is \( 4184 \, \text{J/kg} \cdot \text{°C} \).- Rearrange the formula to find \( \Delta T \): \[ \Delta T = \frac{\Delta KE}{mc} \]Substitute \( m = 13.5 \) kg, and \( \Delta KE \) from Step 3 to find \( \Delta T \).

Solve for Temperature Increase

Now substitute all known values into the equation from Step 4 to find the maximum possible temperature increase of the water, \( \Delta T \).

Key concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. When we talk about a bullet moving through water, it has kinetic energy because it is in motion. The formula for kinetic energy is given by \[ KE = \frac{1}{2} mv^2 \]Where \( m \) is the mass of the object and \( v \) is its velocity. The faster the bullet moves and the heavier it is, the more kinetic energy it has.

In the exercise, the bullet's initial and final kinetic energies are calculated to determine how much energy it loses as it slows down passing through the water. This lost kinetic energy can contribute to increasing the water’s temperature as it transforms from motion energy to thermal energy.

Here’s why it’s important: kinetic energy transformation helps us understand how energy conservation works in systems where different types of energy interact, such as a bullet and water in this case.

• Makes calculations regarding energy transfer easier.
• Helps in predicting results in real-world interactions.
• Assists in visualizing the energy transformation process.

Heat Transfer

Heat transfer is the process by which energy is exchanged between different bodies due to a temperature difference. In the context of our bullet and water example, heat transfer occurs as the lost kinetic energy from the bullet is transferred to the water. This causes a rise in water temperature.

Heat transfer can occur in three ways: conduction, convection, and radiation, but here we're interested in the transformation due to energy already in motion, not the method of heat transfer. The bullet losing energy inside the water is primarily about energy conversion due to mechanical interactions.

• Heat is seen as a form of energy in transit.
• It requires a temperature gradient.
• Usually results in a change in state or temperature of the substance involved.

Understanding this helps us predict how energy loss (like kinetic energy turning into heat) can manifest as changes in material properties, like temperature, explaining indirectly why, in this scenario, the water’s temperature may increase.

Specific Heat Capacity

Specific heat capacity is a property that tells us how much heat is required to change the temperature of a unit mass of a substance by one degree Celsius. It's a unique property for every material and plays a crucial role in determining how a substance reacts under thermal influence.

For water, the specific heat capacity is \[ 4184 \, \text{J/kg} \cdot \text{°C} \]This means that to raise the temperature of 1 kg of water by 1°C, 4184 Joules of energy are required. In the exercise, knowing the specific heat capacity of water allows us to relate the lost kinetic energy of the bullet to the resulting temperature change in the water.

• A high specific heat capacity means the substance requires more energy to change its temperature.
• This concept is essential in calculating the energy transaction in temperature-related problems.
• Gives insights into material properties and usage in thermodynamic processes.

By using the specific heat capacity, we can solve for the temperature increase of the water with the formula \( Q = mc\Delta T \) , rearranged to find \( \Delta T \) once \( Q \) is known.

Temperature Change

Temperature change is the difference in temperature a substance experiences as it absorbs or releases heat. In this problem, we are calculating how much the temperature of water increases as it absorbs the heat from the kinetic energy lost by the bullet.

Through the formula \[ Q = mc\Delta T \]where \( \Delta T \) represents the change in temperature, we can understand how much the temperature of a given mass of water changes based on the heat it gains.

• The formula connects temperature change to heat energy and material properties.
• Allows prediction of physical behavior under specific conditions.
• Essential in thermal management across various applications.

In the provided exercise, we utilized this formula by rearranging it to\[ \Delta T = \frac{\Delta KE}{mc} \]which allowed us to measure the maximum potential increase in water temperature as a result of total energy transfer from the bullet. This concept is pivotal when measuring the impact of an energy source on a substance's temperature.