Problem 93 - University Physics with Modern Physics

Question

Spacecraft Reentry. A spacecraft made of aluminum circles the earth at a speed of 7700 \(\mathrm{m} / \mathrm{s} .\) (a) Find the ratio of its kinetic energy to the energy required to raise its temperature from \(0^{\circ} \mathrm{C}\) to \(600^{\circ} \mathrm{C}\) . (The melting point of aluminum is \(660^{\circ} \mathrm{C}\) . Assume a constant specific heat of 910 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K} .\) (b) Discuss the bearing of your answer on the problem of the reentry of a manned space vehicle into the earth's atmosphere.

Step-by-Step Solution

Answer

The kinetic energy is 54 times greater than the needed thermal energy to reach 600\( ^{\circ} \)C, emphasizing the importance of thermal protection during reentry.

Step 1: Calculate the Kinetic Energy

The kinetic energy (KE) of the spacecraft can be calculated using the formula \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the spacecraft and \( v = 7700 \, \text{m/s} \) is its velocity. Since the mass is not specified, we will assume the mass to be an arbitrary \( m \) and calculate the KE in terms of \( m \): \[ KE = \frac{1}{2}m(7700)^2 = 2.9645 \times 10^{7} m \text{ J} \]

Step 2: Calculate the Thermal Energy Required

The energy required to raise the temperature of the spacecraft from \( 0^{\circ} \text{C} \) to \( 600^{\circ} \text{C} \) can be found by \( Q = mc\Delta T \), where \( c = 910 \, \text{J} / \text{kg} \cdot \text{K} \) is the specific heat capacity and \( \Delta T = 600 \, \text{K} \). Substituting these values gives:\[ Q = m \times 910 \times 600 = 546,000 m \text{ J} \]

Step 3: Determine the Ratio of Kinetic Energy to Thermal Energy

The ratio of the kinetic energy to the thermal energy is found by dividing the kinetic energy by the thermal energy:\[ \text{Ratio} = \frac{KE}{Q} = \frac{2.9645 \times 10^{7} m}{546,000 m} = \frac{2.9645 \times 10^{7}}{546,000} \approx 54.26 \]

Step 4: Discuss the Reentry Implication

The ratio of about 54:1 means the kinetic energy is 54 times greater than the energy required to raise the temperature to near the melting point. This indicates that during reentry, a significant portion of the kinetic energy must be absorbed and dissipated to prevent the spacecraft's temperature from reaching or exceeding the melting point of aluminum, highlighting the critical role of thermal protection systems.

Key Concepts

Understanding Kinetic Energy

Kinetic energy is a fundamental concept in physics associated with the motion of an object. It is given by the equation:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass of the object and \( v \) is its velocity. Kinetic energy increases with both the mass and the square of the velocity of the object. This means that even small increases in velocity can result in large increases in kinetic energy.
For a spacecraft traveling at extremely high speeds, such as 7700 meters per second, its kinetic energy is substantial. The energy is pivotal during reentry, as it needs to be managed to avoid damage to the spacecraft. Understanding kinetic energy helps design protective systems that can handle the energy changes experienced during reentry.

The Role of Specific Heat Capacity

Specific heat capacity is the amount of energy required to raise the temperature of one kilogram of a substance by one degree Celsius (or one Kelvin). In our scenario, aluminum has a specific heat capacity of 910 J/kg·K. The formula for calculating the energy required to change the temperature is:

• \( Q = mc\Delta T \)

This means, to raise the temperature from 0°C to 600°C, it requires a calculated amount of energy based on the mass of aluminum and its specific heat capacity. Specific heat capacity is a crucial factor when considering how much temperature change a spacecraft can endure without structural integrity failure.

Importance of Thermal Protection Systems

Thermal protection systems (TPS) are essential for safeguarding spacecraft during reentry into Earth's atmosphere. Given the enormous kinetic energy that must be dissipated, a TPS is designed to absorb and withstand high levels of heat generated as kinetic energy transitions to thermal energy.
These systems use materials with high melting points and adequate specific heat capacities to minimize the heat reaching critical parts of the spacecraft.

• For instance, ablative materials that burn off and take excess heat with them or insulating materials that reflect heat.

A well-designed TPS ensures the spacecraft doesn't reach the melting point of materials like aluminum, thus preserving its structural integrity.

Understanding Temperature Change in Reentry

The temperature change a spacecraft undergoes during reentry is drastic because of the conversion of kinetic energy into thermal energy.

• This conversion is due to aerodynamic heating as the spacecraft interacts with the Earth's atmosphere.

Using the calculated energy needed to raise the spacecraft to near its melting point, we can appreciate the challenge of managing significant heat increases. Understanding this helps engineers create cooling strategies and select materials that will handle the extreme temperature changes effectively without compromising the spacecraft.
The study of these temperature changes is key in ensuring a safe and successful return to Earth, balancing the kinetic energy transformation into heat and the ability of the spacecraft to dissipate it without overheating.