Lesson Explainer: Kinetic Energy Mathematics • Third Year of Secondary School

In this explainer, we will learn how to calculate the kinetic energy of a moving particle of mass that moves with a velocity of magnitude .

An object that is moving is said to have kinetic energy. The faster the object moves, the more kinetic energy it has.

The kinetic energy of an object can be increased by doing work on it. If a force acts on an object as that object moves along a path, we say that the force does work on the object. If the force also causes the object to accelerate, then the speed of the object will increase, so its kinetic energy will also increase.

Imagine throwing a bowling ball down the lane of a bowling alley. When you throw the ball, you exert a force on it. That force causes the ball to accelerate; hence, it does work on the ball, increasing its kinetic energy.

The kinetic energy of an object can also be decreased by doing work on it. If the force that acts on an object acts in the opposite direction to the motion of the object, the object will decelerate. The speed of the object will decrease, so its kinetic energy will decrease.

Consider an object of mass , which is moving with a velocity . Recall that the speed of an object is defined by the magnitude of its velocity. We often represent this more simply by using to represent speed; hence, .

The kinetic energy of such an object can be defined in the following way.

Looking at the above definition, we can see that the kinetic energy depends linearly on the mass of an object and depends on the square of the speed of the object.

Before moving forward, let us consider how to obtain the simplified formula shown above by using a vector definition of kinetic energy:

For this definition, we can see that we are taking the dot product of the velocity vector with itself. In general, this dot product can be represented as

Here, represents the angle between the two vectors; however, since both vectors are the same, they point in the same direction, and hence :

We now have that taking the dot product of a vector with itself can be found by squaring the magnitude of the vector. In our case, the magnitude of the velocity vector is its speed, and so

Based on this definition, you will notice that we can calculate the kinetic energy of an object when given only its mass and its speed. Even if the direction in which the object is moving is unknown, the kinetic energy can still be found.

It should also be apparent that the kinetic energy is a nonnegative scalar, since both the mass and the speed of an object are themselves nonnegative scalars.

Let us take a look at some examples.

In the next example, we will see that this relationship can be used in reverse. When given the kinetic energy of an object along with the object’s mass or speed, the remaining quantity can be found.

In the previous example, it is worth noting that in order to find an object’s speed using its mass and kinetic energy, we took the square root of some quantity.

Ordinarily, square roots have two valid solutions, both a positive and a negative solution. In order to understand why our answer was not , we must remember that speed is defined as a nonnegative scalar. This means that when finding the speed of an object, as in the method above, we can ignore the negative solution.

For our next examples, we will be working with velocity vectors. Recall that we will now need to use the vector form of the kinetic energy equation:

In practice, velocities will often be given in forms such as

When trying to find the kinetic energy, taking the dot product of the vector with itself can be quickly found using a useful shortcut. The calculation reduces to taking the sum of the squares of the coefficients for each of the perpendicular unit vectors:

The following examples will use this simple shortcut.

When given the kinetic energy and the velocity of an object, we can use our formula to find its mass. As shown in example 2 of this explainer, this is demonstrated with a simple rearrangement of our formula:

This reverse method will be explored in the next example, but before doing this, we should take care to note that the reverse method cannot be used to find the velocity of an object when given its kinetic energy and mass. Let us attempt to rearrange our formula to understand why this is the case:

At this point, it becomes clear that we can only find the magnitude of our velocity (i.e., speed) using this information. Since kinetic energy itself is a scalar quantity, it does not capture any information regarding the direction of movement of the object it describes.

For this reason, we cannot regain the directional information using only and (which are both scalar quantities). Although the magnitude of the velocity can be uniquely defined based on this information (as shown in example 2), there are an infinite number of possibilities for the direction of travel of the object. This means the velocity vector cannot be uniquely defined.

In practice, you may encounter systems with more than one moving object. When working with such systems, it may be helpful to instead use relative velocities.

The formula for kinetic energy behaves as expected when inputting relative velocities. In certain cases, it may even cut down on calculation, as finding the kinetic energy of one moving body relative to another moving body can give a clearer picture of the energy that may be transferred if the two objects were to collide. Let us take a look at one such example.

Finally, here are the key points for the information presented in this explainer.