Translational Kinetic Energy

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Meteor showers are majestic phenomena that occur multiple times a year in our solar system. However, did you know that translational kinetic energy plays a major role in our ability to see them as they fall? When meteors approach the Earth’s atmosphere, gravity attracts them which results in their descent toward the earth at high speeds of 13,000 meters per second. These high speeds, as well as their large mass, means that the meteors have high amounts of kinetic energy. However, atmospheric friction causes some of their kinetic energy to be lost. Consequently, meteors begin to burn up as the lost kinetic energy converts to heat through ionization. This ionization leads to the formation of the long, visible meteor tails we see as linear streaks in the night sky. From our point of view, the shooting stars of meteor showers move in straight lines, meaning that they have translation kinetic energy. Therefore, let this article be a starting point in expanding our knowledge of translational kinetic energy by contrasting its definition to that of rotational kinetic energy and going over relevant examples.

Translational Kinetic Energy, Meteor Shower, StudySmarter Fig. 1: The movement of a meteor shower demonstrates translational kinetic energy.

Review of Key Terms

Before diving into translational kinetic energy, let us review some key terms, starting with the term scalar.

A scalar refers to a quantity with a magnitude but no direction.

Energy is a scalar quantity.

Energy is a system’s ability to do work.

Work measures the amount of energy transferred as a result of an object moving some distance due to an external force.

Energy exists in many forms; however, all energy can be classified as kinetic or potential. Therefore, when calculating energy, the mathematical formula required will depend on the type of energy in question. However, for the purpose of this article, we will only focus on kinetic energy and translational motion.

Kinetic energy is the energy associated with motion.

The word "kinetic" derives from the Greek word "kinētikos" which means "to move."

The SI unit of energy is the joule, denoted by $ \mathrm{J}. $

Translational Kinetic Energy Definition

Having reviewed energy and kinetic energy, let us now define and discuss translational kinetic energy.

Translational Kinetic Energy

Translational kinetic energy is a scalar quantity, which means it has no direction. It only has magnitude.

Translational kinetic energy is the energy due to translational motion.

Translational motion corresponds to a one-dimensional motion along a straight path.

Translational Kinetic Energy Formula

The mathematical formula corresponding to the definition of translational kinetic energy is

\[K_{\mathrm{T}} = \frac{1}{2}mv^2\]

where $m $ is the mass measured in $ \mathrm{kg} $ and $ v $ is the measured velocity in $ \mathrm{\frac{m}{s}}. $ Note that $v^2 = |\vec{v}|^2.$ However, since translational kinetic energy is scalar, vector notation can be suppressed.

Mass and velocity are proportional to translational kinetic energy. If the mass or velocity increases, the translational kinetic energy will increase. Conversely, if the mass or velocity decreases, the translational kinetic energy will decrease.

Translational Kinetic Energy vs. Rotational Kinetic Energy

Before comparing translational and rotational kinetic energy, we must first define rotational kinetic energy and discuss its corresponding formula.

Rotational Kinetic Energy

Rotational kinetic energy is also a scalar quantity, which means it has no direction. Unlike translational kinetic energy, however, its magnitude is determined by the angular speed of an object.

Rotational kinetic energy is energy due to rotational motion.

Rotational motion refers to objects rotating about an axis and is sometimes called angular or circular motion.

The corresponding mathematical formula to the definition of rotational kinetic energy is

\[K_{\mathrm{rot}} = \frac{1}{2}I\omega^2\]

where $ I $ is the moment of inertia measured in $ \mathrm{{kg}\,{m^2}} $ and $ \omega $ is the angular velocity measured in $ \mathrm{\frac{rad}{s}}. $

The moment of inertia is the measurement of an object's resistance to angular acceleration. Formulas involving an object's moment of inertia will vary depending on the shape of the object.

Relationship between Rotational Kinetic Energy and Translational Kinetic Energy

Rotational and translational kinetic energies describe different types of motion, yet their formulas are very similar in form. Why? This is the result of the relationship between linear and rotational motion as they are equivalent counterparts of one another. Linear velocity, $ v $, and angular velocity, $ \omega $, relate to one another by the formulas

\[ v=\omega{r} \]

and

\[\omega = \frac{v}{r}.\]

However, the relationship between mass and inertia is not as straightforward. The rotational analog of mass is called the moment of inertia. An object's moment of inertia describes how mass is distributed relative to the axis of rotation. Meanwhile, mass itself describes the amount of matter within an object. Hence, mass is the measure of an object's resistance to changing its motion. Therefore, the more inertia an object has, the more mass it has.

Total Kinetic Energy of a System

Although systems can contain only translational or rotational kinetic energy, both types can be present in a single system. For example, cars move with translational kinetic energy, while their tires move with translational and rotational kinetic energy. Therefore, the total kinetic energy of the system is the sum of rotational kinetic energy and translational kinetic energy. Its corresponding formula is

\[K_{\mathrm{total}} = K_{\mathrm{T}} + K_{\mathrm{rot}}.\]

Reference Frames and Translational Kinetic Energy

Translational kinetic energy may be measured differently depending on an observer’s reference frame.

A reference frame is an abstract coordinate system from which one can specify the motion and location of bodies relative to an arbitrarily chosen point of origin.

Reference frames are classified as inertial or non-inertial. Inertial reference frames are frames where Newton's first law applies, objects at rest remain at rest, and objects in motion remain in motion. These types of frames can be stationary or move with a constant velocity. A non-inertial reference frame is an accelerating frame, undergoing either linear acceleration or angular acceleration around an axis.

This concept can be hard to understand, so let us complete a thought experiment. Imagine you are riding a train, which moves with constant velocity. You notice a child throwing a ball up into the air. As the train barrels past the next platform, people on the platform see this as well. However, although you each witness the ball thrown in the air at the same time, you have two different reference frames. Your frame of reference is moving while the frame of reference of people on the platform is stationary. Consequently, you witness the ball moving vertically up and down because of gravity. The people on the platform, however, see a parabola where the horizontal part of the ball's velocity is equal to the train's velocity.

Translational Kinetic Energy, Reference Frame, StudySmarter Fig. 2: The motion of an object is viewed differently depending on an observer's frame of reference.

Translational Kinetic Energy Examples

To solve translational kinetic energy problems, one can apply the formula for translational kinetic energy to different problems. As we have defined translational kinetic energy and rotational energy and discussed their relationship, let us work through some examples to gain a better understanding of the concepts. Note that before solving a problem, we must always remember these simple steps:

Read the problem and identify all variables given within the problem.
Determine what the problem is asking and what formulas are needed.
Apply the necessary formulas and solve the problem.
Draw a picture if necessary to provide a visual aid

Examples

Let us apply our new knowledge of translational kinetic energy to the following two examples.

A $ 40\,\mathrm{kg} $ runner moves with a velocity of $ 1.7\,\mathrm{\frac{m}{s}} $. Calculate the translational kinetic energy of the runner.

Translational Kinetic Energy, Running, StudySmarter Fig. 3: Running demonstrates the concept of translational kinetic energy.

After reading the problem, we are given the mass and velocity of the runner,

Therefore, applying the translational kinetic energy formula, our calculations are:

\[\begin{align} K_{\mathrm{T}} &= \frac{1}{2}mv^2\\ &= \frac{1}{2}\left(40\,\mathrm{kg}\right)\left(1.7\,\mathrm{\frac{m}{s}}\right)^2\\&=57.8\,\mathrm{J}.\end{align}\]

The runner has a translational kinetic energy of $ 57.8\,\mathrm{J}. $

Now let's complete a slightly more difficult example.

A $ 7000\,\mathrm{kg} $ train moves with a velocity of $ 42\,\mathrm{\frac{m}{s}} $ while its $12$ wheels rotate with an angular velocity of $ 8.6\,\mathrm{\frac{rad}{s}} $ each. Each wheel have a mass $ 106\,\mathrm{kg} $ and a radius of $ 0.91\,\mathrm{m} $. Calculate the rotational, translational, and total kinetic energy of this system. The moment of inertia for a wheel is $I_{\text{wheel}} = \frac{1}{2}mr^2.$

Translational Kinetic Energy, Moving Train, StudySmarter Fig. 4: A moving train demonstrates the concept of translational kinetic energy.

After reading the problem, we are given the following quantities:

mass of the train and its wheels
velocity of the train
angular velocity of the wheels
radius of the wheels

Therefore, applying the formulas for rotational and translational kinetic energy, our calculations will be as follows:

Translational kinetic energy:

\[\begin{align} K_{\mathrm{T}} &=\frac{1}{2}mv^2\\ &= \mathrm{\frac{1}{2}(7000\,kg)\left(42\,\frac{m}{s}\right)^2}\\&=\mathrm{6,174,000\,\mathrm{J}}.\\\end{align}\]

Rotational Kinetic Energy:

\[K_{\mathrm{rot}} = \frac{1}{2}I\omega^2\]

Before using this equation, we must calculate the moment of inertia of each wheel.

$$\begin{align}I_\mathrm{wheel}& = \frac{1}{2}mr^2\\ &=\mathrm{\frac{1}{2}(106\,kg)(0.91\,m)^2}\\&= 43.89\,\mathrm{kg\,m^2}.\\\end{align}$$

Now, multiply the above value by 12 in order to determine the moment of inertia for the entire system of wheels.

$$\begin{align}I_\mathrm{system}&= (12)(43.89\,\mathrm{kg\,m^2})\\ &= 526.68\,\mathrm{kg\,m^2}.\end{align}$$

Now using the equation for rotational kinetic energy,

$$\begin{align}K_\mathrm{rot}&= \frac{1}{2}I\omega^2\\&= \frac{1}{2}(526.68\,\mathrm{{kg\,m^2}})\left(8.6 \mathrm{\frac{rad}{s}}\right)^2\\&= 19,476.63\,\mathrm{J}.\\\end{align}$$

Therefore, the total kinetic of the system is:

$$\begin{align}K_\mathrm{total}&= K_\mathrm{T}+ K_\mathrm{rot}\\&= 6,174,000\,\mathrm{J} + 19,476.63\,\mathrm{J}\\&= 6.19 \times 10^{6}\,\mathrm{J}.\\\end{align}$$

Translational Kinetic Energy - Key takeaways

Energy is a system’s ability to do work.
Work measures the amount of energy transferred as a result of an object moving some distance due to an external force.
Kinetic energy is energy that is associated with motion and can be written in terms of translational or rotational motion.
Rotational Energy is energy due to rotational motion which is motion associated with objects rotating about an axis.
Translational kinetic energy is energy due to linear motion.
Linear motion is a one-dimensional motion along a straight path.
The formula for rotational kinetic energy is $ K_{rot}=\frac{1}{2}I\omega^2 $.
The formula for translational kinetic energy is $ K_{T}=\frac{1}{2}mv^2 $.
The kinetic energy formulas are of the same form because all quantities associated with linear motion have rotational equivalents.

References

Fig. 1: Meteor Shower (https://www.pexels.com/photo/photo-of-sky-during-sunset-1937687/) by Felipe Helfstein (https://www.pexels.com/@felipe-helfstein-871817/) is licensed by CC0 1.0 Universal (CC0 1.0).
Fig. 2: Reference Frames, StudySmarter Originals
Fig. 3: Running (https://www.pexels.com/photo/woman-with-white-sunvisor-running-40751/) by Pixabay (https://www.pexels.com/@pixabay/) is licensed by CC0 1.0 Universal (CC0 1.0).
Fig. 4: Moving Train (https://www.pexels.com/photo/passing-train-on-the-tracks-1598075/) by James Wheeler (https://www.pexels.com/@souvenirpixels/) is licensed by CC0 1.0 Universal (CC0 1.0).

Frequently Asked Questions about Translational Kinetic Energy

Translational kinetic energy is the energy due to linear motion.

The translational kinetic energy equation is one-half times mass times velocity squared.

Some examples of translational kinetic energy include a meteor shower, a person running, and a moving train.

Yes, mass and velocity are proportional to translational kinetic energy. If mass or velocity increases, translational kinetic energy will increase, and if mass or velocity decreases, translational kinetic energy will decrease.

Translational kinetic energy is associated with linear motion while rotational kinetic energy is associated with rotational motion.

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Flashcards in Translational Kinetic Energy15 Start learning

The SI unit for energy is denoted by what symbol?

J for Joules.

Complete the definition:

____ measures the amount of energy transferred as a result of an object moving some distance due to an external force.

Work.

The energy associated with motion is what type of energy?

Kinetic Energy.

Fill in the Blank:

Energy due to one-dimensional motion along a straight path is known as _____.

Translational kinetic energy.

Fill in the Blank:

Energy due to an object rotating about an axis is ______.

Rotational kinetic energy.

Which of the following is the corresponding mathematical formula for the definition of translational kinetic energy?

$ K=\frac{1}{2}mv^2. $

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Review of Key Terms