Micah Zarin's Blog

We begin with the (untrue) assumptions that the laws of physics will not change if we shift our viewpoint in space, shift it in time, or rotate ourselves in any direction. By “untrue,” I mean that these assumptions fail at the quantum level (when you go really really fast, or get really really small). But if we brush these pathological cases aside for now and stick to our simplified assumptions: there is no special place in the universe, no special moment in time, and no preferred direction in space.

Because of that symmetry, there is nothing to single out a moment when an object should suddenly change its motion, nor is there a special direction or place that would make the object speed up or slow down for no reason. In practical terms, if an object is coasting along at a certain speed, we expect it to keep coasting forever, unless something pokes or prods it. Let x(t) be the object’s position as a function of time t, and let v(t) = dx(t)/dt (which means the object’s velocity is the rate of change of position with respect to time). If there is no external cause for the velocity to change, then v(t) remains the same.

Next, define p(t) = m * v(t), which is the object’s momentum. Because the mass m and the velocity v(t) do not spontaneously change, the product m * v(t) stays constant. Symbolically, dp(t)/dt = 0 (meaning the rate of change of momentum with respect to time is zero) so long as nothing external interferes. However, we do observe in daily life that objects speed up, slow down, or turn when pushed or pulled. Therefore, something must be behind these changes, and that “something” is what we call force.

A force, which we can label F(t), is precisely whatever is responsible for altering an object’s momentum. In math terms, F(t) = dp(t)/dt (meaning force is the rate of change of momentum with respect to time). For constant mass, that boils down to F(t) = m * (dv(t)/dt). In everyday language, force is the reason an object’s speed or direction changes. Whether it is a push on a rolling ball, a gravitational pull on a falling apple, or the thrust of a rocket, each example of force is an outside influence that takes advantage of the fact that there is otherwise no preferred state for the object to shift into on its own.

When we solve actual problems—like figuring out why a car accelerates or how a baseball curves through the air—we identify every force (such as friction, gravity, air resistance, tension in a rope, and so on) and sum them. The sum tells us dp(t)/dt, and from there we can predict the object’s position and velocity at any future time. That is the core practical use of the concept of force: it connects the pushes and pulls we apply to the changes in motion that we observe.