# A-level Mathematics/Advanced/Basic Mechanics/Newton's Laws of Motion

## Newton's Laws of Motion

There are 3 laws of motion.

Newton's 1st law of motion states that a body will remain at rest or move with uniform motion in a straight line unless acted upon by an external force.

For example: A bicycle moving at constant speed round a circle is acted upon by a force since it is not moving in a straight line. This example frequently catches people out as the speed may be constant, but the velocity (speed plus direction which is a vector (link)) does change. This law is counter-intuitive, as generally we expect that if we start something moving then it will slow down over time and eventually stop. This is because of Friction (link), which is the external force in this case.

The equations arising from this can be expressed as :

1. d = s*t

(distance = speed * time) assuming constant speed

2. v= u+a*t

where v is final velocity

u is initial velocity

a is acceleration

3. v2 = u2 + 2a*s

Newton's 2nd law on motion states that the force acting on a body is directly proportional to the rate of change of momentum and takes place in the direction of the force. Its mathematical formulation is $F = \frac{dp}{dt}$, where

F: Force in Newtons (N)

t: Time in seconds (s)

p: Momentum in kilogrammes metres per second squared (kg m s-2)

In most cases, we deal only with situations where the mass concerned does not change, such that $F = \frac{dp}{dt} = \frac{d}{dt}(mv) = m\frac{dv}{dt} = ma$. However, if, like in most other books, F = ma is used so much as to create a habit, it can be disastrous. Hence, it is recommended to remember the p form or mv form instead of the ma form, unless the user is not comfortable with math.

Now that force is defined, we can compare the concept of force, impulse and instantaneous and average velocity. Force and instantaneous velocity are time derivatives of momentum and displacement respectively, while impulse and average velocity is about longer time spans than that.

$F = \frac{dp}{dt}$, $I = \frac{\Delta p}{\Delta t}$, $v = \frac{dx}{dt}$, $\mbox{average } v = \frac{\Delta x}{\Delta t}$

While average velocity still has its uses in A'levels, impulse has practically no use in the A'level curriculum. However, impulse is a useful construct to use when dealing with varying forces, for example in collisions. Nonetheless, $\Delta mv$ is still a viable replacement for impulse.

Newton's 3rd law of motion states that if a body A exerts a force on body B then, Body B will exert an equal but opposite force on body A.

This is originally stated as 'For every action there exists an equal and opposite reaction'. Specifically if a cat is sitting on the pavement then the cat is pushing down exactly as hard as the pavement is pushing up. This must be true as otherwise we'd have either a cat flying into the air (pavement pushing up harder) or a cat shaped dent in the pavement (cat pushing down harder).

This law, commonly used to equate known forces with unknown forces and derive the unknown ones, is of paramount importance. Useful with blocks on slanted surfaces in my experience.

These laws are re-stated (in a slightly more formal style) in Newtonian Physics.