Newton’s Laws of Motion

Today we have a simple post which will break down Newton’s laws of motion so even the layman can understand them inside out (hopefully). Mathematical formula will be introduced, though no previous knowledge of mathematics is required to follow this through – i’m going to explain every single piece of notation as best I can. For those that have a degree of mathematically proficiency already I do apologise that this may seem far too easy! For those that don’t, grab a pen and paper and stick with this!

Newton’s laws of motions are three laws that together form the basis of classical mechanics. Newton’s insight was so advanced for his day (17th century) and it was he who taught us for the first time how to understand the spatial behaviour of objects and the relationship between forces and motion. Physics may have advanced leaps and bounds since Newton’s day but the three laws remain solid pillars of the classical world.


The First Law:

The first law states that if the sum of all the forces acting on an object is zero, then the velocity of the object is constant. [Note constant, not necessarily zero.] i.e if no force acts upon an object it continues to do whatever it was doing before, if it was at rest it continues to sit still, if it was moving it continues to move without changing its velocity. This is known as enacting uniform motion. Myth bust: let’s just clarify what velocity is -velocity is a quantity that incorporates the speed and direction of a moving object. The speed of an object (say 5 miles/hour) is the same regardless if the object is moving in a straight line or a circle. However if the object is moving at 5 miles/hour but in a circle it is changing direction and hence its velocity is not constant. Now let’s get some notation involved with this first law:

To represent force we use the symbol F. Now this sign:  means summation and in front of the F would mean a sum of all the forces in question. So this first expression below means the sum of all the forces on the object add to 0. Now the position of an object is represented in physics by the symbol x. The rate at which an object changes its position over time is its velocity, represented by the symbol v (commonly thought of as speed but we know the difference now). How do we represent this rate of change over time? Well there are two common ways: either by writing d(insert quantity of interest)/dt (where t is time), the d represents the ‘change in’ so this effectively reads ‘change in quantity of interest over change in time’. Or it is given by writing the quantity of interest with a little dot over the top.

So back to our first law. If there are no forces acting on the object the velocity of the object is constant, or in other words, the rate of change of the velocity is zero. [It does not change, it stays the same/constant!]


There we have it, the first law in all its simple glory.

The Second Law:

The second law states that the rate of change of momentum of an object is proportional to the force applied. Let’s recap what momentum is, momentum is a quantity which combines the mass and velocity of an object. A momentum of a tiny person running very fast is roughly the same as a fat person walking slow. You can get an idea of how different objects would rank with their momentum by thinking of which you would least like to be hit by! You definitely wouldn’t like to be hit by a fast moving car because not only is it heavy (large mass) but its velocity is very high. Momentum is represented by the symbol and is equal to the mass of an object times its velocity. (p = mv) Now to recap the initial statement, you would need to apply a greater force in order to produce a greater change in the momentum of an object – this is intuitive, if it were the other way around our world would seem very strange indeed.

So as used before, d(quantity of interest)/dt indicates the rate of change of that quantity and in this case we are talking about momentum – so we’ll stick in a p or an mv, whichever you prefer. This expression is equal to our Force F. Therefore what this law is saying is that you need a force to act on upon an object in order to change its momentum.


Final step: how does the product mass x velocity (mv) change with time? Well Newton’s laws are valid only for constant mass systems so the m can sort of be ignored when it comes to the rate of change. We take the m out of the d(mv)/dt bracket so we now have md(v)/dt as seen below. What, however, is the rate of change of velocity? Acceleration! Acceleration is the rate of change of velocity of an object how much it speeds up or speeds down (deceleration). The symbol for acceleration is a. Therefore dv/dt can be replaced with simply a. 


So there we have Newton’s second law in all its glory. In a nutshell: a net force applied to an object of mass m, produces an acceleration a. 

If we know the force on and mass of the object we can work out the acceleration. If we know the mass of the object and acceleration it is experiences we can work out the force. And finally if we know the force we’re applying on an object and can measure how much it is accelerating we can deduce its mass! Hurrah!

(Why are some letters typed in bold you may ask? The letters typed in bold represent vector quantities in mathematics – a quantity that has direction as well as magnitude)

The Third Law:

The third law states that for every force that exists there is an equal and opposite force pushing back. If an object say you, exert a force a second object (you push a door) then the second object exerts a force back on you with equal magnitude (a.k.a strength) yet in the opposite direction. If we call the two objects A and B then the notation for this force is simply: FA = −F

In this set up force exerted by object A is referred to as the ‘action’ and the force exerted back by object B the ‘reaction’. This is why sometimes Newton’s third law is referred to as the action-reaction law. This law gives as the insight into forces to understand they are all fundamentally interactions between different objects or bodies. There is no such thing as a force without the presence of its equal and opposite partner. This third law is essentially what allows us to get anywhere. Think about it – when you walk you push back against the floor and the floors pushes you forward. When you swim you push the water backward with your arms and the water simultaneously pushes you forward. The tires of a car push against the road and the road pushes back on the tires, driving the car forward.

Has anybody seen the new sci-fi film Passengers? I do love a good sci-film (especially one set in space) the plot line was a little lack-lustre and quite frankly disturbing in its premise at parts but this isn’t a film review blog so i’ll stop. Anyway there’s a scene near the end of the film where one the main characters is floating in space but being pulled towards the fuel-buring backside of the spaceship. He has a heavy object in his hands and cleverly remembers Newton’s third law to get him out of the sticky situation. By throwing this object towards the fire he himself is propelled backwards in the opposite direction due to the reaction force and manages to return to the safety of the ship. It pays off knowing your basic physics, you never know what situation you might find yourself in…


Anyway that’s all for today – I hope this post was educational without being too taxing! Any questions, post them below!


32 responses to “Newton’s Laws of Motion

  1. I would input that an object moving on a specific axis is angular velocity. The fact that you can have a velocity due any direction on our planet in itself follows to travel in a circle. Even some theories depict that our universe has a certain curvature at least space itself has a curvature when large objects contain enough mass to alter the curvature of space. Perhaps velocity and angular velocity depict different ideas, but in all reality, everything carrying a velocity may very well have an angular velocity as well.


    • Thank you for your thoughts yoshillpjuae – I suppose the distinction would depend on your frame of reference and the axes you choose to define in your coordinate system


  2. Such an informative blog which is also entertaining at the same time.

    “You can get an idea of how different objects would rank with their momentum by thinking of which you would least like to be hit by!”

    My favorite part of this post that made me chuckle 🙂

    Liked by 2 people

  3. Just a few questions if you have time. It appears to me that the earth orbiting the sun has a constant force on it yet it travels at a constant speed should it not accelerate according to Newton?
    In theory if we drop a stone it will accelerate for ever so why could not particles reach the speed of light?
    If we push an object in space it will go on forever at whatever speed it leaves the hand is that correct?
    All objects travel with the sun around the black hole at the centre of the galaxy so even in space there is a force acting on everything in the galaxy ?? Surely this also means nothing travels in a straight line ?
    If a rocket had sufficient fuel it could accelerate indefinitely especially as it becomes lighter as the fuel is used.


    • You are quite right Kersten, it does have a constant force which gives it its acceleration. The Earth rotates in a rough circle – it is always changing direction and therefore undergoing acceleration – the subtlety is defining acceleration.

      In the stone example, it wouldn’t be possible to reach the speed of light due to the trade off between energy and mass – however if you did have some infinite gravitational field in the absence of resistive forces it’s theoretically possible it would go very fast. In practice, you have resistive forces (usually air) which are proportional to velocity, so eventually balance out with the gravitational force giving a constant (terminal) velocity. This is basically what the particle accelerators try to do – lower resistive forces and accelerate particles close to the speed of light.

      With regard to things travelling in straight lines let us not forget reference frames. It is entirely possible to set a reference frame for straight line motion on Earth which is in itself rotating around the sun, hence accelerating.

      Liked by 3 people

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  5. As I understand it a large mass, like a planet, bends space itself into a kind of four dimensional cone so the straightest line for a satellite is the largest elliptical path that its velocity permits. (a circle is one form of ellipse) The orbiting body is continuing to fall towards the planet but its original velocity counters that fall with an attempt to fly from the planet in a straight line the balance of the two motions, towards and away from the planet forms the orbit of the satellite.

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  10. Remembering Newton’s Third Law for the next time I am in space and am being pulled towards the fuel-burning backside of a spaceship.

    Thanks for explaining it in a way I can understand, Mekhi! I really enjoyed it!


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