Why does a football fly through the air when being kicked? It's because the foot exerts a force on the football! Forces determine how objects move. Therefore, to make calculations and predictions about the trajectory of any object we need to understand the relationship between forces and motion. Sir Isaac Newton noticed this and came up with three laws that summarize the effects that force has on the motion of an object. That is right; with only three laws, we can describe all motion. Their accuracy is so good that this was enough to calculate the trajectories and interactions that allow us to walk on the moon! The first law explains why objects cannot move on their own. The second is used to calculate the motion of projectiles and vehicles. The third one explains why guns recoil after shooting and why the combustion with the expelling of gasses results in an upward thrust for a rocket. Let's go through these laws of motion in detail and explore how they can be used to explain the world we see around us by looking at some real-life examples.
Forces and motion: Definition
In order to develop a good understanding of how forces and motion are related, we will need to get familiar with some terminology, so let's start by explaining what we refer to as motion and force in more detail.
We say that an object is in motion if it is moving. If it is not moving, we say that it is in repose.
The specific value of the velocity at a given time defines the state of motion of an object.
Force is any influence that can cause a change in the state of motion of an object.
A force can be thought of as a push or pull that acts on an object.
Forces and motion properties
It is very important to keep in mind that velocity and forces are vectors. This means that we need to specify their magnitude and direction to define them.
Let's consider an example where we can see the importance of the vector nature of velocity to talk about the state of motion of an object.
A car is heading west at a constant speed of![]()
. After an hour, it turns and continues at the same speed, heading north.
The car is always in motion. However, its state of motion changes even if its speed remains the same the whole time because, at first, it is moving to the west, but it ends up moving to the north.
A force is also a vector quantity, so it doesn't make sense to talk about forces and motion if we do not specify its direction and magnitude. But before going into this in more detail, let's talk about units of force. The SI units of force are newtons![]()
. One newton can be defined as a force that produces an acceleration of one meter per second squared in an object with a mass of one kilogram.
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Forces are usually represented by the symbol![]()
. We can have many forces acting on the same object, so next, we will talk about the basics of dealing with multiple forces.
Force and motion basics
As we will see later, forces determine the motion of objects. Therefore, to predict the motion of an object, it is very important to know how to deal with multiple forces. Since forces are vector quantities, they can be added together by adding their magnitudes based on their directions. The sum of a group of forces is called the resultant or net force.
The resultant force or net force is a single force that has the same effect on an object as two or more independent forces acting on it.
Fig. 1 - To calculate the resultant force, all the forces acting on an object must be added as vectors
Have a look at the above image. If two forces act in opposite directions, then the resultant force vector will be the difference between them, acting in the direction of the force with greater magnitude. Conversely, if two forces act in the same direction, we can add their magnitudes to find a resultant force that acts in the same direction as them. In the case of the red box, the resultant force is![]()
towards the right. On the other hand, for the blue box, the resultant is![]()
towards the right.
While talking about sums of forces, it is a good idea to introduce what unbalanced and balanced forces are.
If the resultant of all the forces acting on an object is zero, then they are called balanced forces and we say that the object is in equilibrium.
As the forces cancel each other, this is equivalent to having no force acting on the object at all.
If the resultant is not equal to zero, we have an unbalanced force.
You will see why it is important to make this distinction in the later sections. Now let's continue by looking at the relation between forces and motion through Newton's laws.
Relationship between forces and motion: Newton's Laws of Motion
We mentioned previously, that forces can change the state of motion of an object, but we haven't said exactly how this happens. Sir Isaac Newton formulated three fundamental laws of motion that describe the relationship between the motion of an object and the forces acting on it.
Newton's first law of motion: Law of inertia
Newton's First Law
An object continues to be in a state of rest or move with uniform velocity until an external unbalanced force acts on it.
This is closely related to an inherent property of every object with mass, called inertia.
The tendency of an object to keep moving or preserve its state of rest is called inertia.
Let us look at an example of Newton's First Law in a real-life.
Fig. 2 - Inertia causes you to keep moving when a car suddenly stops
Imagine that you're a passenger in a car. The car is moving in a straight line when, suddenly, the driver makes an abrupt stop. You get thrown forward even if nothing pushes you! This is the inertia of your body resisting a change to its state of motion, trying to keep moving forward in a straight line. According to Newton's first law, your body tends to maintain its state of motion and resist to the change - slowing down - imposed by the braking car. Luckily, wearing a seat belt can stop you from being thrown forward abruptly in the case of such an event!
But what about an object originally at rest? What can this inertia principle tell us in that case? Let us look at another example.
Fig. 3 - The football remains at rest because no unbalanced force is acting on it
Look at the football in the above image. The ball remains at rest as long as there is no external force acting on it. However, if someone exerts force by kicking it, the ball changes its state of motion - stops being at rest - and begins to move.
Fig. 4 - When the ball is kicked, a force acts on it for a short time. This unbalanced force makes the ball leave the rest, and after the force is applied, the ball tends to continue moving with constant velocity
But wait, the law also says that the ball will continue to move unless a force stops it. However, we see that a moving ball eventually comes to rest after being kicked. Is this a contradiction? No, this happens because there are multiple forces such as air resistance and friction that act against the motion of the ball. These forces ultimately cause it to stop. In the absence of these forces, the ball will continue to move with constant velocity.
From the above example, we see that an unbalanced force is necessary to produce motion or change it. Keep in mind that balanced forces are equivalent to having no force acting at all! It doesn't how many forces are acting. If they are balanced, they won't affect the state of motion of the system. But how exactly does an unbalanced force affect the motion of an object? Can we measure this? Well, Newton's second law of motion is all about this.
Newton's second law of motion: Law of mass and acceleration
Newton's Second Law
The acceleration produced in an object is directly proportional to the force acting on it and inversely proportional to the mass of the object.
Fig. 5 - The acceleration caused by a force is directly proportional to the force but inversely proportional to the mass of the object
The above image illustrates Newton's Second Law. Since acceleration produced is directly proportional to the force applied, doubling the force applied to the same mass causes the acceleration to double as well, as shown in (b). On the other hand, since acceleration is also inversely proportional to the mass of the object, doubling the mass while applying the same force causes the acceleration to be reduced by half, as shown in (c).
Remember that velocity is a vector quantity that has a magnitude - speed - and a direction. Since acceleration occurs whenever velocity changes, a force producing an acceleration on an object can:
- Change both the speed and direction of the motion. For example, a baseball hit by a bat changes its speed and direction.
Change the speed while the direction remains constant. For example, a car braking keeps moving in the same direction but slower.
Change the direction while the speed remains constant. For example, the earth moves around the sun in a motion that can be considered circular. While it is moving at approximately the same speed, its direction is constantly changing. This is because it is subject to the gravitational force of the sun. The following pictures show this using a green arrow to represent the earth's velocity.
Fig. 6 - Earth moves approximately at the same speed, but its direction constantly changes due to the sun's gravitational force, describing an approximately circular path
Force and motion formula
Newton's second law can be mathematically represented as follows:
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Note that if multiple forces are acting on the body, we have to add them to find the resultant force and then the acceleration of the object.
Newton's second law is also very often written as![]()
. This equation states that the net force acting on a body is the product of its mass and acceleration. The acceleration will be in the direction of the force that is acting on the body. We can see that the mass appearing in the equation determines how much force is needed to cause certain acceleration. In other words, the mass tells us how easy or difficult it is to accelerate an object. Since inertia is the property of a body resisting a change in its motion, mass is related to inertia, and it is somehow a measure of it. This is why the mass appearing in the equation is known as inertial mass.
Inertial mass quantifies how difficult it is to accelerate an object and it is defined as the ratio of the applied force applied to the produced acceleration.
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We are now ready for the final Law of Motion.
Newton's Third Law of Motion: Law of action and reaction
Newton's Third Law of Motion
Every action has an equal and opposite reaction. When one body exerts a force on another (action force), the second body responds by exerting an equivalent force in the opposite direction (reaction force).
Note that the action and reaction forces are always acting on different bodies.
Fig. 7 - By Newton's third law, when a hammer hits a nail, the hammer exerts a force over the nail but the nail also exerts an equal force on the hammer in the opposite direction
Consider a carpenter hammering a nail into a floorboard. Let's say that hammer is being driven with a force of magnitude![]()
. Let us consider this as the action force. For the small interval that the hammer and the nail are in contact, the nail responds by exerting an equal and opposite reaction force![]()
on the head of the hammer.
What about the interaction between the nail and the floorboard? You guessed it! When the nail strikes, exerting a force on the floorboard, the floorboard exerts a reaction force on the tip of the nail. Therefore, when considering the system nail-floorboard, the action force is exerted by the nail and the reaction by the floorboard.
Force and motion examples
We have already seen some examples showing how force and motion are related while introducing Newton's laws. In this last section, we will see some examples of force and motion in everyday life.
It is very intuitive to think that something in repose will keep in repose unless a force acts on it. But remember that Newton's First Law also says that an object in motion remains in the same state of motion - same speed and same direction - unless a force changes this. Consider an asteroid moving through space. Since there is no air to stop it, it continues moving at the same speed and in the same direction.
And as mentioned at the beginning of the article, a rocket is a great example of Newton's third law, where the expelled gases have a reaction force on the rocket, producing a thrust.
Fig. 8 - The gases expelled by the rocket and the thrust are an example of an action-reaction pair of forces
Let's look at a final example and try to identify all the laws of motion that are applicable to the situation.
Consider a book lying on a table. Which laws of motion do you think are being applied here? Let's go through all of them together. Even though the book is at rest, there are two forces at play.
- The weight of the book pulls it down against the table.
- By Newton's third law, there is a reaction from the table to this weight, acting on the book. This is called the normal force.
Fig. 9 - The table responds to the weight of the book pressing against it by exerting a normal force
When an object interacts with another by making contact with it, the second object generates a reaction force perpendicular to its surface. These forces, perpendicular to the interacting objects' surfaces, are called normal forces.
Normal forces are called that way not because they are 'common' but because 'normal' is another way to say perpendicular in geometry.
Returning to our example, since the forces acting on the book are balanced, the resultant force is zero. This is why the book remains at rest, and there is no motion. If now, an external force pushed the book to the right, according to Newton's Second Law, it would accelerate in this direction because this new force is unbalanced.
Fig. 10 - The book remains at rest because no unbalanced force is acting on it
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