Question

At sea level, a nitrogen molecule in the air has an average kinetic energy of \(6.2 \cdot 10^{-21}\) J. Its mass is \(4.7 \cdot 10^{-26} \mathrm{~kg}\). If the molecule could shoot straight up without colliding with other molecules, how high would it rise? What percentage of the Earth's radius is this height? What is the molecule's initial speed? (Assume that you can use \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\); although we'll see in Chapter 12 that this assumption may not be justified for this situation.

Short answer

Also, what is its initial speed? Answer: The nitrogen molecule would rise approximately 1.346 x 10^5 m high, representing about 0.02114% of the Earth's radius. Its initial speed is approximately 391.62 m/s.

Step-by-step solution

Find the initial velocity of the molecule

We can find the initial velocity (v) using the kinetic energy formula: \(KE = \frac{1}{2} m \cdot v^2\) Where KE is the kinetic energy, m is the mass, and v is the velocity of the molecule. Solving for v, we get: \(v = \sqrt{\frac{2 \cdot KE}{m}}\) Now, let's plug in the given values for KE and m: \(v = \sqrt{\frac{2 \cdot (6.2 \cdot 10^{-21} \mathrm{~J})}{(4.7 \cdot 10^{-26} \mathrm{~kg})}}\)

Calculate the initial velocity

By plugging in the values, we can now solve for v: \(v \approx \sqrt{\frac{2 \cdot (6.2 \cdot 10^{-21})}{(4.7 \cdot 10^{-26})}} \approx 391.62 \mathrm{~m}/\mathrm{s}\) The initial velocity of the nitrogen molecule is approximately 391.62 m/s.

Conservation of energy

We will use the conservation of energy principle, which states that the total mechanical energy (the sum of kinetic and potential energy) remains constant in the absence of any external forces. Let's call the maximum height h: At the starting point (sea level): KE = \(6.2 \cdot 10^{-21}\) J, PE = 0 At maximum height: KE = 0, PE = mgh Since the total energy remains constant, we can equate the sum of the kinetic and potential energy at the beginning to the sum at the maximum height of the molecule: \(KE_{initial} + PE_{initial} = KE_{final} + PE_{final}\)

Calculate the height (h)

By plugging in the known values and solving for h, we get: \((6.2 \cdot 10^{-21} \mathrm{~J}) = 0 + (4.7 \cdot 10^{-26} \mathrm{~kg}) \cdot (9.81 \mathrm{~m}/\mathrm{s^2}) \cdot h\) Solve for h: \(h \approx \frac{(6.2 \cdot 10^{-21})}{(4.7 \cdot 10^{-26}) \cdot 9.81} \approx 1.346 \cdot 10^{5} \mathrm{~m}\) The nitrogen molecule would rise up to approximately 1.346 x 10^5 m high.

Calculate the percentage of the Earth's radius

The Earth's radius is approximately 6,371,000 meters. To find the percentage of the Earth's radius of this height, we can calculate the ratio between the maximum height and the Earth's radius: \(percentage \approx \frac{1.346 \cdot 10^{5} \mathrm{~m}}{6,371,000 \mathrm{~m}} \cdot 100\) Now let's calculate the percentage: \(percentage \approx \frac{1.346 \cdot 10^{5}}{6,371,000} \cdot 100 \approx 2.114 \cdot 10^{-2}\%\) This height represents approximately 0.02114% of the Earth's radius. Summary: - The height the nitrogen molecule would rise is approximately 1.346 x 10^5 m. - This height represents about 0.02114% of the Earth's radius. - The initial speed of the molecule is approximately 391.62 m/s.

Key concepts

Conservation of Energy

The principle of conservation of energy is an essential concept in physics. It states that the total energy in a closed system remains constant if no external forces act upon it.
This means that energy can neither be created nor destroyed; it can only be transformed from one form to another.
In the context of our exercise, we're dealing with mechanical energy, which consists of kinetic energy (KE) and potential energy (PE).

Initially, a nitrogen molecule at sea level has a certain amount of kinetic energy and no potential energy, as it is at the reference height of zero.
As the molecule rises, its height increases, converting kinetic energy into potential energy.
At the maximum height, all the kinetic energy is converted into potential energy.

• Initial point: KE = 6.2 x 10^{-21} J, PE = 0 J
• Maximum height: KE = 0 J, PE = mgh

By conserving the total mechanical energy, we equate the initial kinetic energy with the potential energy at the highest point. This enables us to calculate how high the molecule would rise before its energy runs out and it starts to fall back down.

Potential Energy

Potential energy (PE) is the energy stored by an object due to its position relative to a reference point, often in a gravitational field.
It is given by the formula: \( PE = mgh \), where:

• \(m\) is the mass of the object,
• \(g\) is the acceleration due to gravity (9.81 m/s² at Earth's surface),
• \(h\) is the height of the object above the reference point.

This form of energy plays a crucial role in processes involving vertical motion.
As an object gains height, its potential energy increases while kinetic energy decreases.
In our example, a nitrogen molecule reaches a point where all its initial kinetic energy is converted to potential energy. By applying the formula for potential energy, we solve for \(h\), demonstrating how high the molecule can rise given its initial energy at sea level.

Earth's Radius

Understanding Earth's radius helps contextualize the scale of distances and heights in scientific calculations.
The average radius of Earth is approximately 6,371 kilometers or 6,371,000 meters.
Knowing the Earth's radius allows us to compare vertical distances to more familiar measurements of planetary scale.

In the exercise, calculating the height to which the nitrogen molecule would rise provides insight into how small this height is relative to Earth's size.
We determined that the height is about 134,600 meters, or approximately 0.02114% of Earth's radius.
This helps us appreciate the sheer size of Earth compared to the relatively short distances molecules travel in such scenarios.

Understanding these measurements can broaden our perspective on the vastness of the planet, enhancing not only scientific comprehension but also enriching our appreciation of space and scale.