# How to calculate kinetic energy

You may have heard the expression “energy is not created nor destroyed; it is transformed”. This simple but powerful law is the basis of all physical phenomena and refers to processes we witness every day. For example, when you drive a car, the chemical energy in the fuel is converted by the engine into rotational energy, which is then transferred to the wheels using a drive shaft.

Another example is hydroelectric energy production: in a dam, water is collected at a certain height and then left to fall freely to a lower height while passing through a turbine, which converts water’s potential energy into electrical energy. This energy is then used in your house to power different home appliances, such as a washing machine, a microwave, or an electric stove. Here, electrical energy is converted into rotational, electromagnetic, and thermal energies, respectively.

All these types of energies can be categorized into two main groups: potential and kinetic energies. The latter are related to movement and are usually applied in the context of rigid moving objects. For example, a moving car has a certain kinetic energy, which is the result of many energy transformations starting from burning the fuel. A heavy rock falling down a cliff also has a certain kinetic energy, which is the result of its gravitational potential energy being continuously transformed into movement.

Let’s learn what exactly is kinetic energy, how it relates to moving objects and how to calculate it.

## How to calculate kinetic energy

To calculate the kinetic energy of a moving object, follow these simple steps:

1. Measure the body’s mass, m, in kilograms.
2. Determine the body’s speed, v, in m/s. This value equals the magnitude of its velocity vector at the desired moment at which the kinetic energy will be determined.
3. Use the following equation to calculate the body’s kinetic energy in joules:

## What is kinetic energy?

When we talk about moving objects, we tend to relate their motion to a certain force. For example, if you kick a ball resting on the ground, you will probably say it moves because a force was applied to it by your foot. Here, Newton’s second law of motion comes in handy, since it directly relates the acceleration (a) experienced by any body of mass m, when a force (F) acts on it:

When you see a falling object, for instance, an apple falling from a tree, you might relate its movement to the force of gravity, which acts on all bodies in virtue of their mass. The difference between these two examples is that, while your foot needs to come in contact with the ball to effectively make it move, gravity can act on any mass without having to touch it.

In our first example, a force acts on the ball while your foot touches it. According to equation 2, the ball will accelerate depending both on its mass and on how hard you kick it. Nevertheless, once the ball moves away from your foot and is no longer in contact with it, the force you apply on it disappears. In this case, equation 2 indicates the ball will not accelerate further. However, if you do this experiment, you will observe the ball will keep moving for a while. Let’s try and understand this further:

Newton’s second law of motion describes how and why objects accelerate, meaning, how their velocity changes with time due to forces. These changes can refer to both a magnitude and a direction change of the velocity vector. The first alter the object’s speed, while the latter only change their trajectory. If you are not sure about these concepts, go ahead and read our article on how to calculate acceleration.

We have established that, in our previous example, the ball will not accelerate further after it has been kicked. This means, it will tend to move in a straight line at constant speed. According to the definition of acceleration, if the ball does not change its direction of movement and keeps a constant speed, its acceleration will be zero. Nevertheless, something has changed: the ball, which was originally at rest, is now moving.

If you do this experiment on a soccer field, you will probably notice the ball will eventually stop moving after rolling for some distance. This is due to another force acting in the direction opposite to the ball’s movement: friction. This corresponds to the horizontal component of the reaction force being applied by the ground to the ball, as the following free body diagram shows:

Here, the normal force is opposite to the ball’s weight, and the force of the kick, Fk, is opposite to friction, Ff. In this scenario, the ball keeps rolling after departing from your foot because it has gained energy from the kick. This initial energy would be “stored” in the ball forever if no friction was present —as in space—, but, since friction against the ground exists, it is slowly converted into another form of energy: heat. You may observe a similar phenomenon when rubbing your hands together: after doing it for a while your palms get hot due to the friction between both surfaces. Since the ball continuously loses energy to friction, it moves slower and slower, until it eventually stops.

Thinking about motion in terms of energy and energy transfers, instead of considering the forces that make objects move or stop moving, can be very helpful sometimes. When you kick a ball, for example, energy is transferred from your foot to the ball. Since energy cannot be destroyed, it stays “within” the ball, until it is either transferred to another body or converted into another type of energy. This is why the ball keeps moving in a straight line even if you don’t continue exerting a force on it.

Now, you might wonder, what type of energy does the ball, and for that matter, any type of body store, when moving? This is called kinetic energy. Its name comes from the Greek kinein, which means “to move”. This is a type of energy bodies possess due to their movement. Put in simple words: if a body is moving, it contains a certain amount of kinetic energy.

When fuel inside a rocket is burnt, the explosion releases the chemical energy stored in it and transfers a part of it to the body of the rocket in form of kinetic energy. The rest is released as heat, a type of thermal energy. When riding a bicycle, your muscles convey kinetic energy to the pedals by contracting and relaxing, which is possible because your body’s cells release energy from sugars and other substances.

## Types of kinetic energy

We know now that kinetic energy exists in a body if it is moving. Macroscopic objects move in two different ways: translating and rotating. The first refers to the body’s center of mass changing its position over time. A clear example of this is kicking a soccer ball: its center of mass, which is probably located somewhere near its center, will displace in three-dimensional space once you kick it. The other type of movement, rotation, occurs when an object revolves around its center of mass. In our previous example, the soccer ball will also probably rotate while displacing across the field. In this case, the ball exhibits both translation and rotation.

Kinetic energy refers to the energy contained by an object in virtue of its movement. Since macroscopic objects exhibit two types of movement, namely translation and rotation, kinetic energy is also differentiated into translational kinetic energy and rotational kinetic energy

## How to measure kinetic energy?

As we have seen so far, kinetic energy relates to a body’s movement. The faster it moves, the higher the kinetic energy it should contain. If the object is then subjected to friction, its speed will decrease, as will its kinetic energy.

There seems to be a relationship between the speed of a moving body and its kinetic energy. Since any change in an object’s speed is related to the appearance of an acceleration, we can use Newton’s second law of motion to describe kinetic energy. Let’s consider a simple case where a body of mass m is moving along a straight line with an initial speed vi and no friction. If we want to make the object move faster or slower, we need to apply a force to it, thus creating an acceleration, as shown by equation 2.

Imagine the body is moving to the right. If you want to increase its speed, you will have to push it to the right. When you do so, the force you apply increases the kinetic energy of the body until you stop exerting it, thus increasing its speed to a final value vf. Let’s suppose you applied said force for a period in which the body displaced a total distance s, as the following image shows.

According to basic mechanics, the final speed of the object depends on its initial speed, the acceleration created by the force you applied and the total length along which it accelerated:

Where a is the body’s acceleration (which is assumed constant) and s the magnitude of its displacement. If we solve for the acceleration:

Since this acceleration is caused by a force acting on the body, we may use equation 2 to relate both quantities:

This equation may also be written as:

To explain this result, we need to make use of a very important concept in mechanics: work. It is defined as the energy transferred by a force to a body in virtue of the displacement that is generated by it. In our example, when the object is pushed to be accelerated, it displaces a certain distance, which means work is being performed on it. By definition, work is equal to the applied force times the object’s displacement: W = F × s. Therefore, equation 6 can be written as:

This equation indicates that the total work performed upon a moving object is directly proportional to the difference between its final and initial speeds. This makes sense if we consider that the greater the difference between both speeds, the greater had to be the work that generated said acceleration.

The terms including the initial and final velocities are defined as the initial and final kinetic energies of the body, respectively. Since energy is not created nor destroyed, any change from the initial to the final kinetic energy of the body, which in turn implies there was a change of speed —given mass is assumed constant—, indicates work had to be performed.

Conversely, the final kinetic energy of a body equals the work needed to make it move at its final speed starting from rest, when the initial speed is zero. The same applies when slowing down a moving body: work will have to be performed on it to decrease its speed, using a force pointing in the direction opposite to the body’s movement. In this case, since work would be negative, the correct expression is that the body performs work on the object that is slowing it down, for example, your hand. In general, the kinetic energy is defined as:

The units for the kinetic energy are joules (N × m). Since the term for the speed is squared, it is implied that K will always be positive. In turn, if K = 0, the body’s speed is zero, meaning it does not move at all.

If the difference between the final and the initial kinetic energy is positive, work will be positive, which means it was performed on the moving object in order to increase its total movement. In the opposite case, if the difference between the final and initial kinetic energy is negative, it means work was performed by the object onto another object or mean (such as the ground or the air surrounding it), thus decreasing its total movement.

Finally, equation 7 can be written as following, which is known as the work-energy theorem

As we mentioned before, kinetic energy is divided into two main types: translational and rotational kinetic energy. In the second case, this quantity results from the addition of the kinetic energies of all the particles that make up a rigid body which is rotating around its center of mass. In this case:

Where I is the body’s moment of inertia and ω its angular velocity. For bodies exhibiting both rotation and translation, the total kinetic energy is given by the sum of the rotational and translational kinetic energies:

Use the Energy Skate Park simulator by PhET to observe how the potential, kinetic, and thermal energies of a skater are transformed from one into the other when rolling on a ramp. Set the skater on either side of the ramp using your mouse and let him roll down. Make sure to tick the “Bar graph” option to see how the different energies vary in time.

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