In this lecture, the following are introduced:
• Velocity and speed
• Momentum as the quantity of motion
• Newton's laws of motion
• Internal and external forces

Motion

Speed is obviously a part of motion as it describes how quickly something is moving. Now since the direction of movement will also be important, we need to define the related vector quantity called "velocity".
Velocity is defined as speed in a given direction
To represent this vector, a unit length vector in the required direction is taken and multiplied by its scalar speed.
This is written as

In a cartesian coordinate system this is shown as:

The bold face, , indicates that velocity is a vector.
The normal face, , indicates that its size (or magnitude) is a scalar.
The bold face with a "hat", , indicates the direction is given by a vector of unit length.
The notation thus says, that velocity is the speed in the direction of the unit vector.

Newton

The poet Alexander Pope, who lived in Newton's time, wrote:
Nature and Nature's laws lay hid in night
God said:"Let Newton be!" and all was light.

Newton's laws of motion

When the question is asked as to how much motion a body has, it has to be considered that two bodies moving with the same velocity have different amounts of motion because they can produce different effects.

Imagination experiment

Imagine a ping-pong ball and a fire-extinguisher, each falling from 1m onto a thin horizontal glass window.
The light ping-pong ball will bounce off but the heavy fire-extinguisher will smash the glass.
Since they both have the same velocity when they hit the glass, it must be the different mass that causes the different effect. This demonstrates that the motion is different between the two objects and the amount of motion they each have depends on both the mass and the speed in some way. Newton defined it as a simple product.

Momentum

Momentum is defined to be the quantity of motion that a body has, and is equal to the product of its mass and velocity. Newton actually called it, the "quality" of motion, but "quantity" or "amount" is what was meant. In symbols:

Momentum is given the symbol p, and has the unit of kg.m.s^-1. Remember that it is important to look at the symbols but think the words.

Law in Science

It is not modern scientific custom to speak of a "Law of Nature", because the essential nature of law is that it can be broken. The regularities or patterns observed, i.e. the "paradigms", of nature that Science describes are not in principle breakable, so are not "laws", arbitrarily established by some lawmaker. However most textbooks seem to be happy to write of Newton's "laws" of motion.

Newton's first law of motion.

Bodies remain in a state of rest, or of constant speed in a straight line, unless compelled to change by a push or a pull.
In other words, if things don't change they'll probably remain the same!
Less colloquially, with no resultant force being applied to a system, the momentum remains constant.
We will come back to this important principle later, but the converse of this leads to…

Newton's second law of motion

The size of the force on a system is measured by how quickly its momentum changes.

This teaches us how to catch fast moving cricket balls; how to jump off tables; etc.
The secret of reducing the force that you feel is to increase the time during which the momentum changes. If you fell a distance of 1m and landed with rigid knees then you would break both legs. It is important to avoid this, so we have learned from experience to bend at the knees and increase the time during which the collision with the ground happens. When catching a fast moving cricket ball, you need "soft" hands, which means that you let the hands move with the ball for a time. This increases the time to change the momentum and reduces the force experienced and, hopefully, prevents the ball from bouncing out of your hands.

A Russian schoolteacher, Konstantin Tsiolkovsky (1857-1935), investigated what would happen to Newton's law if the mass changed. He became the father of modern rocketry.

However we will look at situations where the mass remains constant, so we end up with:

i.e.

The "force" here is the resultant force acting on the mass.
The unit of force is the Newton. 1 Newton gives a body of mass 1 kg an acceleration of 1 m.s^-2.

Newton's third law of motion

This is about the origin of force produced by contact.
To every action there is an equal and opposite action, i.e. an equal reaction.

It is often stated, "to every action there is an equal and opposite reaction", but this is a double negative - OK, for Shakespeare, but not for modern language. Actually we might anthropomorphise it by, "If you push me, I'll push you back".

It would be interesting if this law was not true, because we could sail yachts by the crew blowing on the sails.

Internal and external forces

Imagination experiment: The farmer and the horse.
A farmer hitches his horse to a cart and orders the horse to pull. The horse says "I have just learnt from Newton's third law, that any force I apply will create an equal but opposite force. If I pull on the cart (red arrow below) then the cart will pull back on me with equal force (mauve arrow). Two equal forces in opposite directions will cancel, so however hard I try it will not move".

The farmer tries a different tack by saying "but you are actually pushing on the ground". The horse replies "same argument, the force that I push against the ground (green arrow below) will be exactly matched by the force of the ground on me (brown arrow). Again there are equal forces in opposite directions which will cancel, so again I cannot move".

The farmer has accepted false logic. In mathematics you can only equate the same kinds of things, otherwise you end up with an argument like this.
½ full glass = ½ empty glass
Multiply both sides by 2
full glass = empty glass, i.e. 1 = 0.
Fullness and emptiness cannot be equated like that. They are different kinds of things.

Internal and external forces

The difficulty with the horse and cart example is that the forces act on different things. The red arrow acts on the cart, the mauve arrow acts on the horse, the brown arrow acts on the horse and the green arrow acts on the ground. If you want to know what happens to the cart, then only consider the forces which act on the cart. When the horse and cart are taken to be a single system then the forces are internal and can be added together. In this case, the forces do cancel, which means that horse and cart are locked together.

Unbalanced external forces cause a system to accelerate. When the cart is considered as the system, then the horse supplies an external force (red arrow below) which causes the cart to move.
When the horse is considered as the system, the the ground supplies an external force (brown arrow) and the cart supplies a force (mauve arrow). If the brown arrow is greater than the mauve arrow then there is a resultant external force on the horse which causes the horse to move.

Internal forces are one part of a system acting on another part of the system to keep the system together. They do not cause motion of the system.
Unbalanced forces acting on the system from outside the system cause it to move.

With this in mind, Newton's second law of motion is expressed as:

The force which causes the motion of a system is the resultant external force acting on the mass of a system. The size of the force equals the product of mass and acceleration.

Now using exactly the same ideas about slopes and areas that were used in kinematics to move between graphs, we have the following:

• The instantaneous force is how quickly the momentum changes. It is given by the slope of the blue tangent on the momentum vs time graph at that instant, and written as shown on the right. By finding the slope at each point on the momentum vs time graph, the lower force vs time graph can be constructed. • The effect of force through time is the red area under the force vs time graph, and written as shown on the right. • The effect of force through time (red area below) is to change the momentum (red arrow above). By finding areas between times, the upper momentum vs time graph can be constructed.

• The instantaneous force is given by how quickly the momentum changes.
• The effect of force through time is to change the momentum.

In the following problems, this procedure is used:
1. Draw a simple diagram of the situation.
2. For each mass draw all the forces which act on it.
3. For each mass: write resultant external force equals mass times acceleration.
4. Solve the equation(s) for each of the unknowns.

Example problems

31. A woman of mass 60 kg stands on a weighing machine in a lift.
Find the mass that the machine displays when :
(a) the lift is stationary.
(b) the lift is accelerating upwards at 2 m.s^-2.
(c) the lift is accelerating downwards at 2 m.s^-2.
-----

(a) the lift is stationary.	The mass recorded on the scale equals the normal reaction scaled to kg. i.e.
(b) the lift is accelerating upwards at 2 m.s-2.	The Normal reaction is greater than the weight, to give a resultant upward force.
(c) the lift is accelerating downwards at 2 m.s-2.	The Normal reaction is less than the weight, to give a resultant downward force.

32. A block of mass 2 kg is at rest on a smooth horizontal table on the x-axis of a coordinate system and 10 m from the origin. A constant horizontal force of 5 N acts on the block in the positive x direction. Find
(a) the force of the table on the block. ans 19.6 N up
(b) the acceleration of the block. ans 2.5 m.s^-2
(c) the distance of the block from the origin 4 s after the application of the force.

The table's upward reaction force N equals the downwards weight N = 2g = 2 × 9.8 = 19.6 N Acceleration is the ratio of force to mass a = F ÷ m = 5 ÷ 2 = 2.5 m.s-2 in the positive x direction

Acceleration makes a change in the speed equal to the area under the acceleration vs time graph. Here the acceleration is constant (horizontal line) so the area will be a rectangle i.e. Δv = a × Δt = 2.5×t Since the initial speed is zero (it was at rest) v = 2.5×t The speed increases linearly with time from rest. Speed makes a change in position equal to the area under the speed vs time graph. In this case the area will be triangular i.e. Δx = ½ × 2.5 × 42 = 20 m However the block starts from +10 m, so with the change in position (+20 m), the block will be at 30 m.

33. A block of mass 7 kg is projected upwards along a smooth inclined plane inclined at 30° with an initial speed of 6 m.s^-1. Find
(a) the deceleration of the mass up the plane.
(b) how far up the plane the mass rises before (momentarily) coming to rest.

There are two forces acting on the block, the weight and the normal reaction from the plane.     -->	The weight is seen to act in two right-angled ways. The weight is now completely replaced by one component which attempts to push the block into the plane (Winto) and another component (Wdown) which attempts to push the block down the plane. <--
The reaction force is created from the component of the weight that pushes into the plane (Winto). The size of the reaction force will equal the size of Winto, i.e. N = 7g Cos300, but in the opposite direction. These two forces will be internal forces keeping the block and the plane in contact.    -->	The component of the weight down the plane is the resultant external force on the block. Wdown = 7g Sin300. This will affect the motion of the block by causing it to slow from its initial upward motion. It will produce a negative acceleration, a, where a = - (7g Sin300)÷7 = - 4.9 m.s-2. <--
	The green area gives the change in speed. The speed at any time is given by v = 6 - 4.9t The change in position will be given by the yellow rectangular area [how far it would move if it continued at the inital speed] plus the effect of the acceleration [the triangle bounded by the dotted arrows and the red line], i.e. Δx = 6t + ½×t×(-4.9t) = 6t - 2.45t2 The speed goes to zero when 0 = 6 - 4.9t0 i.e. at t0 = 6÷4.9 = 1.2 s At this time the block will have moved upwards by Δx = 6×1.2 - 2.45×1.22 = 3.6 m

Summarising:

The velocity of a system is its speed in a given direction.  
The momentum of a system is the product of its mass and velocity.  

Newton's laws of motion:
1. Bodies remain in a state of rest, or of constant speed in a straight line, unless compelled to change by a push or a pull.
2. The size of the force on a system is measured by how quickly its momentum changes.  
3. To every action there is an equal and opposite action, i.e. an equal reaction.

Internal forces are one part of a system acting on another part of the system, they do not cause motion, they keep a system together.

The force which causes the motion of a system is the resultant external force.
The effect of resultant external force through time is to change the momentum:

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