We have seen that an object won’t change its velocity unless it is pushed or pulled in some way (post 16.2). Here we will think about what happens when its velocity does change. To keep things simple, we’ll think about an object moving in a straight line, just as we did when we thought about measuring movement (post 16.12). Then “velocity” and “speed” are the same (post 16.2) and the mathematics is simplified.
Suppose an object changes its speed from v to v + Δv in an interval of time Δt. Then we define its average acceleration to be Δv/Δt. Since the units that we use to measure v are m.s-1 (m/s), the units for acceleration are m.s-1/s which becomes m.s-2 (m/s2). If the average acceleration does not change during any increment of time within the time interval Δt then the average acceleration is equal to the true acceleration. We usually call the “true acceleration” simply the acceleration. If you studied physics at school you may have learned equations that relate distance moved, to speed, time and acceleration. If you did, be very careful – they only work for a constant acceleration, which is when the acceleration does not change.
In everyday speech we often use “acceleration” to mean going faster and use the term “deceleration” to mean going slower. In science, we use “acceleration” for both; from the definition of acceleration, going slower is simply a negative acceleration.
According to Newton’s second law (post 16.2), we have to push or pull an object to make it accelerate. This pushing or pulling is called a force (post 16.4). From everyday experience we know that it is easier to push a smaller object, like a bicycle, then a bigger object like a car. We also know that a crash involving a fast car is likely to do more damage than a crash involving a slow car. In the same way, you can put a bullet on a sheet of glass. But if a sheet of glass meets a fast bullet, shot from a gun, it will break, as in the picture below.
This everyday experience is used to define the concept of momentum. Momentum is something called mass, which is a measure of how difficult it is to change the motion of an object, multiplied by velocity. More mathematically, the momentum p of an object is defined by p = mv where m is its mass and v is its velocity. Don’t confuse m, representing mass with the standard abbreviation for a metre, m. That’s why I’m using italics to represent algebraic symbols and ordinary font to represent units. Also note that although m for metres is a standard abbreviation, there are no standard symbols representing concepts like mass, so we have to define such symbols whenever they use them. (If you are familiar with a British television quiz called University Challenge, you will now realise that some of its “science” questions are meaningless).
Notice that we haven’t given a proper definition of “mass” for the same reason that we didn’t define “length” or “time” (post 16.12). So we define the unit of mass which is the kilogrammes (kg). In 1795, the gramme was defined as the mass of a cubic centimetre of water at its melting point. However, we now have a standard kilogramme, in the same way as there used to be a standard metre, which is kept in Paris. As far as I know, unlike the standards of length and time, it has yet to be replaced by anything more reliable, although this has been suggested. You might expect that we would continue to use a gramme rather than a kilogramme (a thousand grammes) as the basic unit of mass. Unfortunately, this is a peculiarity of SI units (see post 16.12 to see what this means).
Since momentum is a mass multiplied by a velocity, the units we use to measure it are kg.m.s-1 (kg.m/s).
Momentum contains all the factors that influence our ability to change the motion of an object; so we define an average force, F, in terms of the change of momentum, Δp that occurs in a time period, Δt, by F = Δp/Δt. In words, force is the rate of change of momentum. If the mass of an object remains constant Δp = Δ(mv) = mΔv. Then the average force becomes F = Δp/Δt =mΔv/Δt = ma where a represents acceleration. This is the definition of force that most of us learnt in school. It works most of the time but mass may not be constant; as a simple example, think about the mass of a vehicle and its contents as it consumes fuel. If the mass and acceleration don’t change during the time period Δt, then F is a constant force.
You might expect that we would measure force in units of kg.m.s-2 (the units of momentum divided by the units of time). But we give this unit a special name, the newton (N), when it is used to measure a force. If you have followed this blog at all carefully, you will know who it is named after!
So we have now defined several important ideas in terms of three intuitive concepts (mass, length and time) that can’t be defined themselves. However, everything is internally consistent and the main justification is that it works!