KAHIBARO
Table of Contents
Motion Energy of a Moving Object
Translational kinetic energy is the energy an object has because its whole body is moving from one place to another. The word translational means motion of the object as a whole, not spinning or vibrating. If a ball flies through the air, a car moves along a road, or a person runs forward, each has translational kinetic energy.
This quantity helps us connect motion to energy. An object moving faster has more translational kinetic energy. An object with more mass also has more translational kinetic energy if it moves at the same speed.
The Formula
For an object of mass $m$ moving with speed $v$, the translational kinetic energy is
$$K = \frac{1}{2}mv^2$$
Here, $K$ is kinetic energy, $m$ is mass, and $v$ is speed.
Important formula:
$$K = \frac{1}{2}mv^2$$
Translational kinetic energy depends on the square of the speed. If speed doubles, kinetic energy becomes four times larger.
Because speed is squared, even a small increase in speed can produce a much larger increase in kinetic energy.
Why Speed Matters So Strongly
The formula shows that kinetic energy is proportional to $v^2$, not just $v$. This means speed has a very strong effect.
If two objects have the same mass, their kinetic energies compare like this:
| Speed change | New kinetic energy |
|---|---|
| $v \to 2v$ | $4K$ |
| $v \to 3v$ | $9K$ |
| $v \to \frac{1}{2}v$ | $\frac{1}{4}K$ |
If the mass changes instead, the kinetic energy changes directly with mass.
| Mass change | New kinetic energy |
|---|---|
| $m \to 2m$ | $2K$ |
| $m \to 3m$ | $3K$ |
| $m \to \frac{1}{2}m$ | $\frac{1}{2}K$ |
Units
Since kinetic energy is a form of energy, its SI unit is the joule, written as $\mathrm{J}$.
Using the formula,
$$K = \frac{1}{2}mv^2$$
the unit is
$$\mathrm{kg}\cdot \left(\mathrm{m/s}\right)^2 = \mathrm{kg}\cdot \mathrm{m}^2/\mathrm{s}^2 = \mathrm{J}$$
The SI unit of translational kinetic energy is
$$1\ \mathrm{J} = 1\ \mathrm{kg}\cdot \mathrm{m}^2/\mathrm{s}^2
$$
Kinetic Energy is a Scalar
Translational kinetic energy does not have a direction. It is a scalar quantity. Even though velocity has direction, kinetic energy depends on speed, which is the magnitude of velocity.
This means an object moving east at $10\ \mathrm{m/s}$ and another moving west at $10\ \mathrm{m/s}$ have the same translational kinetic energy if their masses are equal.
Zero and Positive Values
Because the formula contains $v^2$, translational kinetic energy can never be negative.
If the object is at rest, then $v = 0$, so
$$K = 0$$
If the object is moving, then $v > 0$, so
$$K > 0$$
Translational kinetic energy is never negative.
$$K \ge 0$$
It is zero only when the object is not moving.
Simple Examples
Consider a $2\ \mathrm{kg}$ object moving at $3\ \mathrm{m/s}$. Its translational kinetic energy is
$$K = \frac{1}{2}(2)(3^2) = 1 \cdot 9 = 9\ \mathrm{J}$$
Now consider a $4\ \mathrm{kg}$ object moving at the same speed:
$$K = \frac{1}{2}(4)(3^2) = 2 \cdot 9 = 18\ \mathrm{J}$$
Doubling the mass doubles the kinetic energy.
If instead the original $2\ \mathrm{kg}$ object moves at $6\ \mathrm{m/s}$, then
$$K = \frac{1}{2}(2)(6^2) = 1 \cdot 36 = 36\ \mathrm{J}$$
Doubling the speed changed the kinetic energy from $9\ \mathrm{J}$ to $36\ \mathrm{J}$, which is four times larger.
Comparing Different Objects
It is often useful to compare kinetic energies without calculating everything from the beginning.
Suppose object A has mass $m_A$ and speed $v_A$, and object B has mass $m_B$ and speed $v_B$. Then
$$\frac{K_A}{K_B} = \frac{m_Av_A^2}{m_Bv_B^2}$$
This makes comparison easier.
For example, if one car has twice the mass of another but the same speed, it has twice the translational kinetic energy. If it has the same mass but twice the speed, it has four times the translational kinetic energy.
Visual Idea
A faster object carries more motion energy. A heavier object also carries more motion energy. The diagram below shows two moving blocks, where the faster or heavier one has greater translational kinetic energy.
Translational Versus Other Kinds of Kinetic Energy
This chapter focuses only on translational kinetic energy, the energy of motion of the whole object from place to place. Some objects can also have rotational kinetic energy if they spin. For example, a rolling wheel can have both translational and rotational kinetic energy. Here we consider only the translational part.
Main Idea to Remember
Translational kinetic energy tells us how much energy an object has because it is moving through space.
Key facts:
$$K = \frac{1}{2}mv^2$$
More mass means more translational kinetic energy.
More speed means much more translational kinetic energy, because speed is squared.
Translational kinetic energy is a scalar and is always nonnegative.