KINETIC ENERGY: CHARACTERISTICS, TYPES, EXAMPLES, EXERCISES - PHYSICAL - 2025

The kinetic energy of an object is that which is associated with its movement, which is why objects at rest lack it, although they may have other types of energy. Both the mass and the speed of the object contribute to the kinetic energy, which in principle, is calculated by the equation: K = ½ mv ²

Where K is the kinetic energy in joules (the unit of energy in the International System), m is the mass, and v is the velocity of the body. Sometimes kinetic energy is also denoted as E _c or T.

Figure 1. Cars in motion have kinetic energy by virtue of their motion. Source: Pixabay.

Characteristics of kinetic energy

-Kinetic energy is a scalar, therefore its value does not depend on the direction or the sense in which the object moves.

-It depends on the square of the speed, which means that by doubling the speed, its kinetic energy does not simply double, but increases 4 times. And if it triples its speed, then the energy is multiplied by nine and so on.

-Kinetic energy is always positive, since both the mass and the square of the velocity and the factor ½ are.

-An object has 0 kinetic energy when it is at rest.

-Many times the change in the kinetic energy of an object is of interest, which can be negative. For example, if at the beginning of its movement the object had greater speed and then began to brake, the _final difference K - _initial K is less than 0.

-If an object does not change its kinetic energy, its speed and mass remain constant.

Types

Regardless of what kind of motion an object has, whenever it moves it will have kinetic energy, whether it moves along a straight line, rotates in a circular orbit or of any other kind, or experiences a combined rotational and translational motion..

In this case, if the object is modeled as a particle, that is, although it has mass, its dimensions are not taken into account, its kinetic energy is ½ mv ², as stated at the beginning.

For example, the kinetic energy of the Earth in its translational movement around the Sun, is calculated knowing that its mass is 6.0 · 10 ²⁴ kg with a speed of 3.0 · 10 ⁴ m / s is:

More examples of kinetic energy will be shown later for various situations, but for now you might wonder about what happens to the kinetic energy of a particle system, since real objects have many.

Kinetic energy of a particle system

When you have a system of particles, the kinetic energy of the system is calculated by adding the respective kinetic energies of each one:

Using the summation notation it remains: K = ½ ∑m _i v _i ², where the subscript “i” denotes the i-th particle of the system in question, one of the many that make up the system.

It should be noted that this expression is valid whether the system is translated or rotated, but in the latter case, the relationship between the linear velocity v and the angular velocity ω can be used and a new expression for K can be found:

In this equation, r _i is the distance between the i-th particle and the axis of rotation, considered fixed.

Now, suppose that the angular velocity of each of these particles is the same, which happens if the distances between them are kept constant, as well as the distance to the axis of rotation. If so, the subscript “i” is not necessary for ω and it comes out of the summation:

Rotational kinetic energy

Calling I to the sum in parentheses, we obtain this other more compact expression, known as rotational kinetic energy:

Here I is called the moment of inertia of the particle system. The moment of inertia depends, as we see, not only on the values ​​of the masses, but also on the distance between them and the axis of rotation.

By virtue of this, a system may find it easier to rotate about one axis than about another. For this reason, knowing the moment of inertia of a system helps to establish what its response will be to rotations.

Figure 2. People spinning on the carousel wheel have rotational kinetic energy. Source: Pixabay.

Examples

Movement is common in the universe, rather it is rare that there are particles at rest. At the microscopic level, matter is composed of molecules and atoms with a certain particular arrangement. But this does not mean that atoms and molecules of any substance at rest are thus also.

In fact, the particles inside the objects vibrate continuously. They do not necessarily move back and forth, but they do experience oscillations. The decrease in temperature goes hand in hand with the decrease of these vibrations, in such a way that the absolute zero would be equivalent to a total cessation.

But absolute zero has not been achieved so far, although some low-temperature laboratories have come very close to achieving it.

Motion is common both on the galactic scale and on the scale of atoms and atomic nuclei, so the range of kinetic energy values ​​is extremely wide. Let's look at some numerical examples:

-A 70 kg person jogging at 3.50 m / s has a kinetic energy of 428.75 J

-During a supernova explosion, particles with kinetic energy of 10 ⁴⁶ J.

-A book that is dropped from a height of 10 centimeters reaches the ground with a kinetic energy equivalent to 1 joule more or less.

-If the person in the first example decides to run at a rate of 8 m / s, his kinetic energy increases until he reaches 2240 J.

-A baseball ball of mass 0.142 kg thrown at 35.8 km / h has a kinetic energy of 91 J.

-On average, the kinetic energy of an air molecule is 6.1 x 10 ^-21 J.

Figure 3. Supernova explosion in the Cigar Galaxy seen by the Hubble telescope. Source: NASA Goddard.

Work theorem - kinetic energy

Work done by a force on an object is capable of changing its motion. And in doing so, the kinetic energy varies, being able to increase or decrease.

If the particle or object goes from point A to point B, the work W _AB required is equal to the difference between the kinetic energy that the object had between point B and that it had at point A:

The symbol "Δ" is read "delta" and symbolizes the difference between a final quantity and an initial quantity. Now let's see the particular cases:

-If the work done on the object is negative, it means that the force opposed the movement. Therefore the kinetic energy decreases.

-In contrast, when the work is positive, it means that the force favored the movement and the kinetic energy increases.

-It may happen that the force does not do work on the object, which does not mean that it is immobile. In such a case the kinetic energy of the body does not change.

When a ball is thrown vertically upward, gravity does negative work during the upward path and the ball slows down, but on the downward path, gravity favors the fall by increasing speed.

Finally, those objects that have uniform rectilinear motion or uniform circular motion do not experience variation in their kinetic energy, since the speed is constant.

Relationship between kinetic energy and moment

The momentum or momentum is a vector denoted P. It should not be confused with the weight of the object, another vector that is often denoted in the same way. The moment is defined as:

P = m. v

Where m is the mass and v is the velocity vector of the body. The magnitude of the moment and the kinetic energy have a certain relationship, since they both depend on the mass and the speed. You can easily find a relationship between the two quantities:

The nice thing about finding a relationship between momentum and kinetic energy, or between momentum and other physical quantities, is that momentum is conserved in many situations, such as during collisions and other complex situations. And this makes it much easier to find a solution to problems of this kind.

Conservation of kinetic energy

The kinetic energy of a system is not always conserved, except in certain cases such as perfectly elastic collisions. Those that take place between almost non-deformable objects like billiard balls and subatomic particles come very close to this ideal.

During a perfectly elastic collision and assuming that the system is isolated, the particles can transfer kinetic energy to each other, but on the condition that the sum of the individual kinetic energies remains constant.

However, in most collisions this is not the case, since a certain amount of the kinetic energy of the system is transformed into heat, deformation or sound energy.

Despite this, the momentum (of the system) is still conserved, because the interaction forces between the objects, while the collision lasts, are much more intense than any external force and under these circumstances, it can be shown that the moment is always conserved.

Exercises

- Exercise 1

A glass vase whose mass is 2.40 kg is dropped from a height of 1.30 m. Calculate its kinetic energy just before reaching the ground, without taking into account air resistance.

Solution

To apply the equation of kinetic energy, it is necessary to know the speed v with which the vase reaches the ground. It is a free fall and the total height h is available, therefore, using the equations of kinematics:

In this equation, g is the value of the acceleration of gravity and v _o is the initial velocity, which in this case is 0 because the vase was dropped, therefore:

You can calculate the square of the velocity with this equation. Note that the velocity itself is not necessary, since K = ½ mv ². You can also plug the velocity squared into the equation for K:

And finally it is evaluated with the data supplied in the statement:

It is interesting to note that in this case, the kinetic energy depends on the height from which the vase is dropped. And just as you might expect, the vase's kinetic energy was on the rise from the moment it began to fall. It is because gravity was doing positive work on the vase, as explained above.

- Exercise 2

A truck whose mass is m = 1 250 kg has a speed of v ₀ = 105 km / h (29.2 m / s). Calculate the work the brakes must do to bring you to a complete stop.

Solution

To solve this exercise, we must use the work-kinetic energy theorem stated above:

The initial kinetic energy is ½ mv _or ² and the final kinetic energy is 0, since the statement says that the truck comes to a complete stop. In such a case, the work that the brakes do is entirely reversed to stop the vehicle. Considering it:

Before substituting the values, they must be expressed in International System units, in order to obtain joules when calculating work:

And so the values ​​are substituted into the equation for the job:

Note that the work is negative, which makes sense because the force of the brakes opposes the movement of the vehicle, causing its kinetic energy to decrease.

- Exercise 3

You have two cars in motion. The former has twice the mass of the latter, but only half its kinetic energy. When both cars increase their speed by 5.0 m / s, their kinetic energies are the same. What were the original speeds of both cars?

Solution

At the beginning, car 1 has kinetic energy K _1o and mass m ₁, while car 2 has kinetic energy K _2o and mass m ₂. It is also known that:

m ₁ = 2m ₂ = 2m

K _1st = ½ K _2nd

With this in mind we write: K _1o = ½ (2m) v ₁ ² and K _2o = ½ mv ₂ ²

It is known that K _1o = ½ K _2o, which means that:

Thus:

Then it says that if the speeds increase to 5 m / s the kinetic energies equal:

½ 2m (v ₁ + 5) ² = ½ m (v ₂ + 5) ² → 2 (v ₁ + 5) ² = (v ₂ + 5) ²

The relationship between both speeds is replaced:

2 (v ₁ + 5) ² = (2v ₁ + 5) ²

Square root is applied to both sides, to solve for v ₁:

√2 (v ₁ + 5) = (2v ₁ + 5)

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill.
2. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 2. Dynamics. Edited by Douglas Figueroa (USB).
3. Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed Prentice Hall.
4. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
5. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 1-2.