Problem 95 - 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 much greater than the thermal energy required, emphasizing the need for heat management during reentry.

Step 1: Calculate Kinetic Energy

The kinetic energy (KE) of the spacecraft is given by the formula: \( KE = \frac{1}{2}mv^2 \). Since we want to find the ratio and the mass \( m \) cancels out, we do not need the actual mass for this ratio. The given speed \( v \) is 7700 m/s, so we have: \( KE = \frac{1}{2}m \times 7700^2 \).

Step 2: Calculate Thermal Energy Required

To raise the temperature of the aluminum from \(0^{\circ} \mathrm{C}\) to \(600^{\circ} \mathrm{C}\), we use the formula for thermal energy \( Q = mc\Delta T \), where \( c \) is the specific heat capacity, \( 910 \, \mathrm{J} / \mathrm{kg} \cdot \mathrm{K} \), and \( \Delta T \) is the change in temperature, \(600^{\circ} \mathrm{C} - 0^{\circ} \mathrm{C} = 600 \, \mathrm{C} \). Thus, \( Q = m \times 910 \times 600 \).

Step 3: Calculate the Ratio

The ratio of kinetic energy to thermal energy needed is \( \frac{KE}{Q} = \frac{\frac{1}{2}m \times 7700^2}{m \times 910 \times 600} \). Simplifying this, the mass \( m \) cancels out, resulting in \( \frac{7700^2}{2 \times 910 \times 600} \). Calculating this gives the ratio.

Step 4: Discussion on Reentry

The calculated ratio will indicate whether the kinetic energy of the spacecraft is significantly higher compared to the thermal energy required for heating up. A higher ratio suggests that a large amount of energy needs to be dissipated to prevent overheating, underlining the importance of thermal protection systems during reentry to safely manage and dissipate kinetic energy as thermal energy.

Key Concepts

Kinetic Energy

Kinetic energy is a form of energy an object possesses due to its motion. When thinking about a spacecraft reentering Earth's atmosphere, kinetic energy becomes a crucial concept because the spacecraft travels at very high speeds.
For any object, including a spacecraft, the kinetic energy (\( KE \)) is calculated using the formula: \( KE = \frac{1}{2}mv^2 \), where:

• \( m \) is the mass of the object
• \( v \) is the velocity of the object

In this scenario, knowing just the speed gives insight into the spacecraft's energy without needing its mass because we're looking at energy ratios. Using the given speed of 7700 meters per second, you can see the kinetic energy is immense, which must be carefully managed during reentry to avoid damaging the spacecraft due to overheating.

Specific Heat Capacity

Specific heat capacity is an important concept when considering how materials will react to changes in temperature. It tells us how much energy is needed to raise the temperature of a given mass of a substance by 1 degree Celsius (or 1 Kelvin).For the spacecraft, which is made of aluminum, the specific heat capacity is given as 910 J/kg·K. This number suggests that aluminum requires 910 Joules of energy to increase the temperature of one kilogram of aluminum by one Kelvin.
Understanding the specific heat capacity helps in calculating the thermal energy required to raise the spacecraft's temperature from 0°C to 600°C. This calculation involves changes in temperature (\( \Delta T \)) and provides insight into ensuring the spacecraft doesn't reach or exceed its melting point, highlighting the importance of thermal regulation.

Thermal Protection Systems

Thermal protection systems (TPS) play a critical role in ensuring that the spacecraft safely reenters Earth's atmosphere. A spacecraft must dissipate its high kinetic energy as thermal energy, preventing overheating which can damage the spacecraft. During reentry, as a spacecraft engages with the atmosphere at high velocity, it faces immense frictional heat. TPS are crafted to handle this by either insulating the spacecraft from extreme thermal conditions or absorbing and dissipating the heat effectively.
Materials used in TPS have high melting points and specific heat capacities to withstand the high temperatures that occur during reentry. Systems must be designed to handle the conversion of kinetic energy into thermal energy in a way that protects the structure and the passengers of the spacecraft. It underlines the critical nature of selecting appropriate materials and designing efficient systems to ensure safety.