Definition of a Force
Since force is the fundamental concept of Dynamics, we must give a clear definition of this concept before we proceed with Newton's Laws. A force is defined (very practically) as a push or a pull. Of course, we experience forces all the time in everyday lives. Whenever we lift something, push something or otherwise manipulate other objects, we are exerting a force. A force is a vector quantity, as it has both a magnitude and a direction. Let us show vector quality of a force practically: when exerting a force, for example pushing a crate, we can change the magnitude of our force by pushing harder or softer. We can also change the direction of our force, as we can push it one way or another. Since a force is a vector, all the rules of vector addition and subtraction, seen in Vectors apply. The vector quality of force allows us to manipulate forces in exactly the same way we manipulated velocity and acceleration in Kinematics.
With a formal definition of force, we can now examine its relation to motion through Newton's laws.
Newton's First Law
So how exactly does a force relate to motion? Intuitively, we can say that a force, at least in some way, causes motion. When I kick a ball, it moves. Newton makes this relation more precise in his first law:
An object moves with constant velocity unless acted upon by a net external force.What does this mean? Let's start by looking at a special case where the constant velocity is zero, i.e. the object is simply at rest. Newton's First Law states that the object will stay at rest unless a force acts upon it. This makes sense: the soccer ball isn't going anywhere unless someone kicks it. This concept is true not only for v = 0, but for any constant velocity. Consider now a ball rolling with a constant velocity. Neglecting friction, the ball will continue to roll with the same velocity until it hits something, or someone kicks it. In physics terminology, it will keep the same velocity until acted upon by a net external force.
What does Newton mean by a net force? Consider a rope being used in a tug of war. There are definitely forces being applied to the rope but, if the two sides pull with the same force, the rope won't move. In this example, the two forces on the rope exactly cancel each other out, and there is no net force on the rope. It is thus possible for forces to act on an object, yet have the net force be zero. When evaluating the motion caused by forces acting on an object, remember to find the vector sum of those forces.
Also included in Newton's First Law, though not explicitly, is the concept of inertia. Inertia is defined as the tendency of an object to remain at a constant velocity. It is a fundamental property of all matter. In a sense, the idea of inertia is unnecessary; it just gives a name to the concept Newton describes in his First Law. However, you're bound to hear the word over and over in physics, so it is important to know to what it refers.
From our concept of inertia, we can develop the idea of an inertial reference frame, meaning a frame in which a body has no observed acceleration. This concept has limited application to classical mechanics, yet is essential for the study of Relativity. Consider a body with no net force acting upon it. For example, imagine yourself in an accelerating automobile. You look out the window, and the ground seems to be accelerating in a direction opposite the motion of the car. Clearly no net forces act upon the ground, yet from the frame of the car the ground is accelerating; in this case the car represents a non-inertial frame, and measurements of inertial fields from non-intertial fields do not conform to the rules of Newton's Laws. If, however, the car is traveling at a constant velocity, the ground will also appear to be moving back with a constant velocity. In this case, the frame of the car is inertial, as no net acceleration is observed. Any inertial reference frame is thus valid one in which to make calculations based on Newton's laws. Before we use these force laws, we must make sure we are making measurements from an inertial frame.