Question

A nitrogen molecule of mass \(m=4.648 \cdot 10^{-26} \mathrm{~kg}\) has a speed of \(300.0 \mathrm{~m} /\mathrm{s}\).

Short answer

Answer: The kinetic energy of the nitrogen molecule is \(2.092 \cdot 10^{-24} \mathrm{J}\).

Step-by-step solution

Write down the given values

We are given the mass \(m=4.648 \cdot 10^{-26} \mathrm{~kg}\) and the speed \(v=300.0 \mathrm{~m/s}\).

Write down the kinetic energy formula

The formula to calculate the kinetic energy is \(KE = \dfrac{1}{2}mv^2\).

Plug in the given values and solve for the kinetic energy.

Now, we just need to plug the given mass and speed into the formula and solve for kinetic energy: \(KE = \dfrac{1}{2}(4.648 \cdot 10^{-26} \mathrm{~kg})(300.0 \mathrm{~m/s})^2\) Calculate the expression inside the parenthesis: \(KE = \dfrac{1}{2}(4.648 \cdot 10^{-26} \mathrm{~kg})(90000 \mathrm{~m^2/s^2})\) Now, multiply the values: \(KE = 2.092 \cdot 10^{-24} \mathrm{J}\)

Write down the final answer

The kinetic energy of the nitrogen molecule is \(KE = 2.092 \cdot 10^{-24} \mathrm{J}\).

Key concepts

Understanding Kinetic Energy Calculation

Kinetic energy is the energy possessed by an object due to its motion. To calculate the kinetic energy of an object, we use the formula:
\( KE = \frac{1}{2}mv^2 \)

In this formula, \( KE \) is the kinetic energy, \( m \) is the mass of the object in kilograms, and \( v \) is the speed of the object in meters per second. The formula shows that the kinetic energy is directly proportional to the mass of the object and the square of its velocity. This means that even a slight increase in velocity will have a significant impact on the kinetic energy, as it is squared in the calculation.

For example, when solving the kinetic energy of a nitrogen molecule, you would substitute the known values of mass and velocity into the formula. By multiplying half of the mass by the square of the velocity, you get the kinetic energy in joules, the standard unit of energy.

The Mass-Velocity Relation in Kinetic Energy

The mass-velocity relation is a key concept in understanding how the mass and velocity of an object affect its kinetic energy. The formula themselves provide insights into this relation:\( KE = \frac{1}{2}mv^2 \)

It's crucial to recognize that while mass and velocity both influence kinetic energy, they do so in different ways. Mass contributes linearly: if you double the mass, the kinetic energy doubles. In contrast, the velocity has a quadratic relationship: if you double the velocity, the kinetic energy increases by a factor of four. This is because the velocity is squared in the kinetic energy formula.This is exceptionally important when examining the kinetic energy of small particles, like the nitrogen molecule in our example. Despite its tiny mass, the relatively high velocity leads to a measurable amount of kinetic energy.

Physics Problem-Solving Techniques

Approaching problems in physics systematically can vastly improve the process of finding solutions. Here are a few steps commonly used in physics problem-solving:

1. Understand and summarize the problem: Identify what is given and what needs to be found.
2. Develop a plan: Determine which physics concepts and formulas are relevant.
3. Execute the plan: Carry out the calculations following the formulas, being careful with units and conversion factors.
4. Review the solution: Check if the answer makes sense in the context of the problem and confirm that the units are correct.

In the case of our kinetic energy problem, we followed these steps by identifying the mass and velocity of the nitrogen molecule, using the kinetic energy formula, calculating with the correct values and units, and finally reviewing our solution to ensure plausibility.