Section 4 - Mechanics - complete

- a scalar has no direction, it's just an amount of something e.g. the mass of a sack of meaty dog food
- a vector has magnitude (size) and direction - like the speed and direction of next door's cat running away

scalars
- mass 
- temperature
- time
- length/distance
- speed
- energy

vectors
- displacement
- velocity & force
- acceleration
- momentum

- start by making a scale drawing of the two vectors (tip-to-tail if they're not already), draw the resultant vector from the tail of the first to the tip of the last, and measure its length and angle

Example
A man walks 3.0m on a bearing of 055^o then 4.0m east. Find the magnitude and direction (to the nearest degree) of his displacement, s

- start by drawing a scale diagram for how far the man walked using a rule and a protractor
- then just measure the missung side with a ruler and the missing angle with a protactor
- s = 6.7 and the angle is equal to 75^o (to the nearest degree)
- so the man's displacement is 6.7m, on a bearing of 075^o 

- when two vectors are at right angles to each other, you can use maths to work it out without a scale drawing

Example
Jemima goes for a walk. She walks 3m north and 4m east. She has walked 7m but she isn't 7m from her starting point. Find the magnitude and direction (to the nearest dregee) of her displacement

- first sketch the vectors tip-to-tail & then draw a line from the tail of the first vector to the tip of the last vector to give you the resultant vector
- because the vectors are at right angles, you get the magnitude of the resultant using pythagoras
- R² = 3² + 4² = 25 so R = 5

- use the triangle to find the angle of her bearing
- use TOA (opposite and adjacent sides)
- tan(theta) = 4/3
- theta = 053^o 

- this is the opposite to finding the resultant & you start from the resultant vector and split it into two components at right angles to each other
- to get the horitzontal component (v_x) use v x cos(theta)
- to get the vertical component (v_y) use v x sin(theta) 
- in these formulae, theta is measured anticlockwise from the horizontal

Example
Charley's amazing floating home is travelling at a speed of 5.0ms^-1 at an angle of 60^o (2 s.f.) up from the horizontal. Find the vertical and horizontal component
- the horizontal component = v x cos theta = 5 cos 60 = 2.5ms^-1 
- the vertical component = v x sin theta = 5 sin 60 = 4.3ms^-1 

- resolving is useful because the two components of a vector don't affect each other
- this means you can deal with the two directtions completely separately

- free body force diagrams show a single body on its own
- the diagram should include all the forces that act on the body, but not the force it exerts on the rest of the world
- forces are vector quantities and so the arrow labels should show the size and direction of the forces
- if a body is in equilibrium (i.e. not accelerating) the forces acting on it will be balanced in each direction
- a body in equailibrium can be at reast or moving with a constant velocity
- coplanar forces are all in the same plane

Examples
- gravity pulls the satellite downwards towards Earth (weight = mxg)
- gravity pulls man down & air resistance pushes man up
- hand pushes toy car forward while friction pulls the car backwards & gravity will pull the car down while the earth pushes the car up

- forces can be in any direction, so they're not always at right angles to each other which can make an awkward calculation
- to make it easier to deal with, you can think of it as two separte forces, acting at right angles to each other
- forces are vectors, so you can use the same method as before

Example
The force F has exactly the same effect as the horizontal and vertical forces F_H and F_v 
Replacing F with F_H and F_V is called resolving the force F

F_H / F = cos(theta) or F_H = Fcos(theta)
and then
F_V / F = sin(theta) or F_V = Fsin(theta)

- when you have three coplanar forces acting on a body in equilibrium, you can draw the forces as a triangle, forming a closed loop
- F₃ is not the sum of F₁ and F₂ as it has to be in the opposite direction to balance the other two forces
- if its a right-angled triangle, you can use pythagoras to find a missing force
- if not, you might have to resolve the forces in each direction

Example
3 teams are taking part in the CGP Fun Day 3-Way Tug of War. If the knot is in equilibrium, find the size of the force P
- P has no vertical component, so you can ignore the vertical components of the other teams as they must cancel each other 
- resolve the horizontal forces:
- (cos41 x 1175) + (cos41 x 1175) - P = 0
- P = 886.7... + 886.7... = 1800N (2 s.f.)

- if two forces act on an object, you find the resultant force by adding the vectors together and creating a closed triangle, with the resultant force represented by the third side
- forces are vectors, so you use vector addition by drawing the forces as vector arrows put 'tip-to-tail' and then you use trig to find the angle and the length of the third side

Example
Two dung beetles roll a dung ball along the ground at constant velocity. Beetle A applies a force of 0.50N northwards while beetle B exerts a force of only 0.20N eastwards. What is the resultant force on the dung ball?
- the resultant force is 0.54N (2 s.f.) at an angle of 22^o (2 s.f.) from north

- the moment of a force depends on the size of the force and how far the force is applied from the turing point:
- momrnt of a force (Nm) =  force (N) x perpendicular distance from the point to the line of action of the force (m)
- M = F x d

- the principle of moments states that for a body to be in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about the same point
Example
Two children sit on a seesaw. An adult balances the seesaw at one end. Find the size and direction of the force that the adult needs to apply.
In equilibrium, total anticlockwise moments = total clockwise moments
so, 400 x 1.5 = (300x1) + 1.5F
600 = 300 + 1.5F
F = (600-300) / 1.5 = 200N downwards

- in a lever, an effor force (in this case from a muscle) acts against a load force (e.g. the weight of your arm) by means of a rigid object (the bone) rotating around a pivot (the joint)
- you can use the principle of moments to answer lever questions

Example
Find the force, E, exerted by the biceps in holding a bag of gold still. The bag of gold weighs 100N and the forearm weighs 20N.
- take moments about A (the elbow bending)
- in equilibrium, total anticlockwise moments = total clockwise moments
- (100 x 0.4) + (20 x 0.2) = 0.04E
- 40 + 4 = 0.04E
- 44 / 0.04 = E = 1100N

- a couple is a pair of forces of equal size which act parallel to each other, but in opposite directions
- the forces are coplanar
- a couple doesn't cause any resultant linear force, but does produce a turing effect (i.e. a moment)
- the size of this moment depends on the size of the forces and the distance between them:
moment of a couple (Nm) = size of one of the forces (N) x perpendicular distance betweem the lines of action of the forces (m)
- M = F x d

Example
A cyclist turns a sharp right corner by applying equal but opposite forces of 20N to the ends of the handlebars. The length of the handlebars if 0.6m. Calculate the moment applied to the handlebars.
- moment = 20 x 0.6 = 12Nm

- the mass of an object is the amount of 'stuff' in it and is measured in kg
- the greater an object's mass, the greater its resistance to a change in velocity (inertia)
- the mass of an object doesn't change if the strength of the gravitational field changes
- weight is a force & its measured in newtons, like all forces
- weight is the force experienced by a mass due to a gravitational field
- the weight of an object does vary according to the size of the gravitational field acting on it
- weight = mass x gravitational field strength (W=mg) where g = 9.81 Nkg^-1 on Earth

Example
Derek the Lion has the same mass on the Earth and the Moon but a different weight.
(name - quantity - earth (g = 9.91 Nkg^-1) - moon (g = 1.6 Nkg^-1)
- mass - mass (scalar) - 150kg - 150kg
- weight - force (vector) - 1470N (3 s.f.) - 240N (2 s.f.)

- the centre of mass of an object is the single point that you can consider its whole weight to act through (whatever its orientation)
- the object will always balance around this point, although in some cases the centre of mass will fall outside the object
- the centre of mass of a uniform, regular solid (e.g. a sphere, a cube) is at its centre

Using Symmetry
- to find the centre of mass for a regular object you can just use symmetry
- the centre of mass of any regular shape is at is centre where the lines of symmetry will cross
- the centre of mass is halfway through the thickness of the object at the point the lines meet

By Experiment
- hang the object freely from a point (e.g. one corner)
- draw a vertical line downwards from the point of suspension
- use a plumb bob to get your line exactly vertical
- hang the object from a different point
- draw another vertical line down
- the centre of mass is where the two lines cross

- an object will topple over if a vertical line drawn downwards from its centre of mass falls outside its base area
- this is because a resultant moment occurs, which provides a turning force
- an object will be stable if it has a low centre of mass and a wide base area
- the higher the centre of mass and the smaller the base area, the less table the object is

- displacement, velocity and acceleration are all vector quantities so the direction is important

- speed - how fast something is moving, regardless of direction
- displacement (s) - how far an object's travelled from its starting point in a given direction
- velocity (v) - the rate of change of an object's displacement (its speed in a given direction)
- acceleration (a) - the rate of change of an object's velocity

- v = change in s / change in t 
- a = change in v / change in t 

- the speed (or velocity) of an object at any given point in time is known as its instantaneous speed (or velocity)
- to find the average speed (or velocity), divide the total distance or displacement by the total time

- a graph of displacement against time for an accelerating object always produces a curve
- if the object is accelerating at a uniform rate, then the rate of change of the gradient will be constant

- a bigger acceleration will have a steeper curve
- a smaller acceleration will have a flatter curve
- a deceleration will result in the line having a decreasing gradient and curving the opposite way 

- when the velocity is constant, the graph's will be a straight line
- velocity = change in displacement (y axis) / change in time (x axis) = gradient
- so velocity on a displacement time graph is just the gradient

- if the gradient isn't constant, it means the object is accelerating
- to find the instantaneous velocity at a certain point, you need to draw a tangent to the curve at that point and find its gradient
- to find the average velocity over a period of time, just divide the total change in displacement by the total change in time
- ti doesn't matter if the graph is curved or not

- acceleration = change in velocity / time taken
- the acceleration is just the gradient of a velocity time graph
- uniform acceleration is always a straight line
- the steeper the gradient, the greater the acceleration
- a curve graph shows changing acceleration
- increasing gradient means increasing acceleration and decreasing gradient means decreasing acceleration or deceleration

- distance travelled = average speed x time so the area under a velocity-time graph tells you the displcement of the object
- the magnitude of this displacement is the distance that object has travelled

Example
A ball is dropped from table-height so it bounces vertically. It bounces twice before someone catches it. The ball's motion while it bounces is shown on the v-t graph. Calculate how high the ball rebounds on the first bounce.
- when the ball is first dropped, the velocity of the ball is negative, so downwards is the negative direction
- the points where the ball hits the floor are shown by the vertical straight lines on the graph, the ball's speed remains roughly the same, but its direction (and velocity) changes the instant it hits the floor
- the point's where the ball's velocity is 0 show where the ball reachers the top of a bounce before starting to fall downwards
- the height of the first bounce is the area under the graph between the time the ball first rebounds from the floor and the time it reaches the top of the bounce
- (3.5 x 0.35) / 2 = 0.61m (2 s.f.)

- an acceleration-time (a/t) graph shows how an object's acceleration changes over time
- the height of the graph gives the object's acceleration at that time
- the area under the graph gives the object's change in velocity
- if a = 0 then the object is moving with constant velocity
- a negative acceleration is a deceleration

Example
The acceleration of a car in a drag race is shown in this acceleration-time graph
a) after how many seconds does the car reach its maximum velocity
- when the acceleration is 0, i.e. after 4 seconds
b) if the car was stationary at t = 0s, calculate its maximum velocity
- change in velocity = area under graph = 0.5*4*25 = 50ms^-1 

- instead of gathering distance and time data using traditional methods, e.g. a stopwatch and ruler, you can be more high tech
- a fairly standard piece of kit you can use for motion experiemnts is an ultrasound position detector
- this is a type of data-logger that automatically records the distance of an object from the sensor several times a second
- if you attach one of these detectors to a computer with graph-drawing software, you can get real-time displacement-time and velocity-time graphs

- the main advantages of data-loggers over traditional methods are:
~ the data is more accurate as you don't have to allow for human reaction times
~ automatic systems have a much higher sampling rate than humans as most ultrasound position detectors can take a reading ten times every second
~ you can see the data displayed in real time

- uniform means constant and there are 4 main equations that are used when solving problems involving uniform acceleration
1 - acceleration is the rate of change of velocity
- therefore a = (v-u) / t 
- so v = u + at
- where u = initial velocity, a = accleration, v = final velocity and t = time taken

2 - s = average velocity x time
- if acceleration is constant, the average velocity is just the average of the intial and final velocities
- so, s = ((u+v) / 2) x t 
- where s = displacement

3 - if you substitute the expression for v from equation 1 into equation 2, you get:
- s = ((u + u + at) x t) / 2 
- s = (2ut + at²) / 2
- s = ut + 1/2at²

4 - use equation 1 in the form a = (v-u)/t
- multiply both sides by s, where s = ((u+v)/2) x t AND this gives you: as = ((v-u)/t) x (((u+v)xt)/2)
- the t's on the right cancel, so: 2as = (v-u)(v+u) and 2as = v² - uv + uv - u² 
- so v² = u² + 2as

- free fall is defined as the motion of an object undergoing an acceleration of 'g'
- acceleration is a vector quantity and acts vertically downwards
- the magnitude of 'g' is usually taken as 9.81ms^-2, although it varies slightly at different points of the Earth's surface
- the only force acting on an object in free fall is its weight
- objects can have an initial velocity in any direction and still undergo free fall as long as the force providing the initial velocity is no longer acting

- for over 1000 years, people believed that heavier objects would fall towards the ground quicker than lighter objects
- Galileo believed that all objects fall at the same rate
- the problem in trying to prove this was that free-falling objects fell too quickly for him to be able to take any accurate measurements and air resistance affects the rate at which objects fall
- Galileo measured the time a ball took to roll down a smooth groove in an inclined plane
- he killed two birds with one stone by rolling it down a plane, which slow's the ball's fall as well as reducing the effect of air resistance
- by rolling the ball along different fractions of the total length of the slope, he found that the distance the ball travelled was proportional to the square of the time taken meaning that the ball was accelerating at a constant rate
- in the end, Newton brought it all together to show and explain why all free falling objects have he same acceleration
- he was able to mathematically show that all objects are attracted towards the Earth due to a force he called gravity

1) set up the equipment (electromagnet, ball bearing attached to it, a switch and a trapdoor)
2) measure the height h from the bottom of the ball bearing to the trapdoor
3) flick the switch to simultaneously start the timer and disconnect the electromagnet, releasing the ball bearing
4) the ball bearing falls, knocking the trapdoor down and breaking the circuit which stops the timer 
- record the time t shown on the timer
5) repeat this experiment three times and average the time taken to fall from this height
- repeat this experiment but drop the ball from several different heights
6) you can then use these results to find g using a graph

- having a small and heavy ball bearning means you can assume air resistance is so small you can ignore it

- having a computer automatically release and time the ball-bearing's fall can meaure times with a smaller uncertainty than if you tried to drop the ball and time the fall using a stopwatch

- the most significant source of error in this experiment will be in the measurement of h
- using a ruler, you'll have an uncertainty of about +/- 1mm
- this dwarfs any error from switch delay or air resistace

1) use your data from the experiment on the last page to plot a graph of height (s) against the time it takes the ball to fall, squared (t²) 
- then draw a line of best fit 
2) you know that with constant acceleratio, s = ut + 1/2at² 
- if you drop the ball, initial speed u = 0, so s = 1/2at²
3) rearranging this gives 1/2a = s/t² or 1/2g = s/t² 
4) so the gradient of the line of best fit (change in s divided by change in t²) is equal to 1/2g
5) g = 2 x (change in s / change in t²) = 2 x (0.44/0.09) = 9.81ms^-2 

- as g is a constant acceleration, you can use it in the equations of motion
- however, because g acts downwards, you need to be careful about directions
- in most cases, upwards is positive and downwards is negative

1) No Initial Velocity (just falling)
- initial velocity u = 0 
- acceleration a =g = -9.81ms^-2 so the motions of equation become
v = gt, v² = 2gs, s = 1/2gt², s = vt/2

2) An initial velocity upwards (thrown up)
- the equations of motion are just as normal, but with a = g = -9.81ms^-2

3) An initial velocity downwards (thrown down) 
Example
Alex throws a stone downwards from the top of a cliff. She throws it with a downwards velocity of 2.0ms^-1. It takes 3.0s to reach the water below. How high is the cliff?
s = ? u = - 2.0 v = ? a = g = -9.81 t = 3.0
s = ut + 1/2gt² = (-2.0 x 3.0) + (1/2 x -9.81 x 3²) = -50.145m 
the cliff is 50m (2 s.f.) high

Sharon fires a scale model of a TV talent show host with a velocity of 100ms^-1 from 1.5m above the ground. How long does it take to hit the ground, and how far does it travel horizontally? Assume the model acts as a particle, the ground is horizontal and there is no air resistance.

Vertical Motion
- its a constant acceleration under gravity
- u = 0, s = -1.5m and a = g
- t = the square root of 2s/g = 0.55300... s
- the model hits the gound after 0.55 seconds

Horizontal motion
- the horizontal motion isn't affected by gravity or any other speed, so it moves at a constant speed
- this means you can use the equation speed = distance/time
- now, v_H = 100ms^-1, t = 0.55300...s and a = 0
- you need to find s_H where v_H is the horizontal velocity and s_H is the horizontal distance travelled rather than the height fallen
- s_H = v_H x t = 100 x 0.55300... = 55m (2 s.f.)

- if something is projected at an angle (e.g. a javelin) you start off with both horizontal and vertical velocity
Method: 
1) resolve the initial velocity into horizontal and vertical components
2) use the vertical component to work out how long it's in the air for and/or how high it goes
3) use the horizontal component to work out how far it goes horizontally while its in the air

Example
a cannonball is fired from ground height at an angle of exactly 40^o with an initial velocity of 15ms^-1 calculate how far the cannonball travels before it hits the ground. assume no air resistance.

1) horizontal component u_H = cos40 x 15 = 11.49...ms^-1 
vertical component u_V = sin40 x 15 = 9.64...ms^-1 

2) - halfway through the balls flight, its v_v will be 0
- u_v = 9.64..., a = -9.81, t = ?
- use v_v = u_v + at = 0 = 9.64... + (-9.81 x t) -----> t = 9.64.... / 9.81 = 0.9
- so the time it takes for it to reach the ground again = 2 x 0.98.... = 1.96...s

3) - there is no horizontal acceleration, so u_H = v_H = 11.49...ms^-1 
- distance = constant speed x time = 11.49... x 1.96... = 22.58... = 23m (2 s.f.)

- newtons 1st law of motion states that the velocity of an object will not change unless a resultant force acts on it
- this means a body will stay still or move in a straight line at a constant speed, unless there's a resultant force acting on it

example
- an appple sitting on a table won't go anywhere because the forces on it are balanced
- reaction (R) (force of the table pushing apple up) = weight (mg) (force of gravity pulling apple down)

- if the forces aren't balanced, the overall resultant force will make the body accelerate
- this could be a change in direction, speed or both

- resultant force (N) = mass (kg) x acceleration (ms^-2)
- F = ma

- it says that the more force you have acting on a certain mass, the more acceleration you get
- it says that for a given force the more mass you have, the less acceleration you get

- consider two balls dropped at the same time, ball 1 being heavy and ball 2 being light
- if you use newtons 2nd law of motion you can see why they will both fall at the same rate

Ball 1
mass = m₁ 
force = F₁ 
acceleration = a₁ 
- by newtons 2nd law, F₁ = m₁ x a₁ 
- ignoring air resistance, the only force acting on the ball is weight, given by W₁ = m₁g (where g = gravitational field strength = 9.81 Nkg^-1) 
- so: F₁ = m₁a₁ = W₁ = m₁g meaning m₁a₁ = m₁g and therfore a₁ = g

Ball 2
mass = m₂ 
force = F₂
acceleration = a₂
- by newtons 2nd law, F₂ = m₂ x a₂ 
- ignoring air resistance, the only force acting on the ball is weight, given by W₂ = m₂g (where g = gravitational field strength = 9.81 Nkg^-1) 
- so: F₂ = m₂a₂ = W₂ = m₂g meaning m₂a₂ = m₂g and therfore a₂ = g

- if an object A exerts a force on object B, then object B exerts and equal but opposite force on object A
- the two forces represent the same interaction but from two different perspectives

1) if you push against a wall, the wall will push back against you just as hard
- as soon as you stop pushing, so does the wall
2) if you pull a cart, whatever force you exert on the rope, the rope exerts the exact oppositr pull on you (unless the rope is stretching)
3) when you go swimming, you push back against the water with your arms and legs, and the water pushes you forwards with an equal-sized force

- newtons 3rd law applies in all situations and to all types of force
- but the pairs of forces are always the same type
- e.g. both gravitational or both electrical

- there are two main types of friction - dry friction between solid surfaces and fluid friction (known as drag, fluid resistance or air resistance)

Fluid Friction or Drag
- 'fluid' is a word that means something that can flow (e.g. a liquid or gas)
- the force depends on the thickness or viscosity of the fluid
- it increases as the speed increases
- it also depends on the shape of the object moving through it, the larger the area pushing against the fluid, the greater the resistance force
- a projectile is slowed down by air resistance
- if you calculate how far a projectile will travel without thinking about air resistance, your answer will be too large

- frictional forces always act in to opposite direction to the motion of the object
- they can never speed things up or start something moving
- they convert kinetic energy into heat and sound

- lift is an upwards force on an object moving through a fluid
- it happens when the shape of an object causes the fluid flowing over it to change direction
- the force acts perpendicular to the direction the fluid flows in

example
cross-section of a plane wing moving through air
- as the wing moves through the air, it pushes down on the air and changes its direction
- this causes an equal and opposite reaction force on the wind

- you will reach a terminal (maximum) speed at some point, if you have a driving force that stays constant and a frictional/drag force that increases with speed

- there are three main stages to reaching terminal speed:
1) the car accelerates from rest, using a constant driving force
2) as the speed increases, the frictional forces increase 
- this reduces the resultant force on the car and hence reduces its acceleration
3) eventually the car reaches a speed at which the frictional forces are equal to the driving force
- there is now no resultant force and no acceleration, so the car carries on at constant speed

- there are two main ways of increasing a vehicle's maximum speed:
1) increasing the driving force, e.g. by increasing the engine size
2) reducing the frictional force, e.g. by making the body more streamlined

- when something's falling through air, the weight of the object is a constant force accelerating the object downwards
- air resistance is a frictional force opposing this motion, which increases with speed
- so before a parachutist opens the parachute, exactly the same thing happens as with the car example

1) a skydiver leaves a plane and will accelerate until the air resistance equals their weight
2) they will then be travelling at a terminal speed
BUT the terminal speed of a person in free fall is too great to land without dying so the parachute is deployed to increase the air resistance and slow them down to a lower terminal speed
3) before reaching the ground he will open his parachute, which immediately increases the air resistance so it is now bigger than his weight
4) this slows him down until his speed has dropped enough for the air resistance to be equal to his weight again
- this new terminal speed is small enough for him to land safely

- the momentum of an object depends on two things, its mass and velocity
- the product of these two values is the momentum of the object
- momentum = mass x velocity

- assuming no external forces act, momentum is always conserved
- this means the total momentum of two objects before they collide equals the total momentum after the collision
- this is useful for working out the velocity of objects after a collision
example
momentum before = momentum after
(75x4) + (5x0) = 125v
300 = 125v
so v = 2.4ms^-1 

- the same principle can be applied in explosions
- e.g. if you fire an air rifle, the forward momentum gained by the pellet equals the backward momentum of the rifle, and you feel the rifle recoiling into your shoulder
example
momentum before = momentum after 
0 = (0.0050 x 200) + (4 x v)
= 1 + 4v
v = -0.25ms^-1 

- an elastic collision is one where momentum is conserved and kinetic energy is conserved i.e. no energy is dissipated as heat, sound etc.
- if a collision is inelastic it means that some of the kinetic energy is converted into other forms during the collision but momentum is always conserved

example
momentum before = momentum after
(2x3) + (0.8x2) = (2x2.6) + (0.8v)
7.6 = 5.2 + 0.8v
2.4 = 0.8v
v = 3ms^-1 

KE before = 1/2mv² (lorry) + 1/2mv² (car)
(1/2 x 2 x 3²) + (1/2 x 0.8 x 2²)
= 11J (2 s.f.)

KE after = (1/2 x 2 x 2.6²) + (1/2 x 0.8 x 3²)
= 10J (2 s.f.)

- the difference in the two values is the amount of kinetic energy dissipated as heat or sound, or in damaginf the vehicles so this is an inelastic collision

- newtons second law says force = rate of change of momentum (mv-mu)/t
- rearranging newtons 2nd law give Ft = mv - mu where Ft is impluse (Ns)
- impulse is the area under a force-time graph

example
the graph shows the resultant force acting on a toy car. if the car is initially at rest, what is its momentum after 3 seconds?
- impulse = change in momentum = mv - mu
- the initial momentum is 0 so impulse mv
- impulse is the area under the graph, so to find the momentum of the car after 3 seconds, you need to find the area under the graph between 0 and 3 seconds
- momentum = area under graph = (15x3)/2 = 22.5 Ns

- the force of an impact is increased by reducing the impact time e.g. the less time your foot is in contact with a football when kicking it, the more force you will kick it with assuming the change in momentum is the same 
- the force of an impact can be reduced by increasing the time of impact e.g. using vehicle safety features
- in order to design vehicles ethically, manufacturers need to make sure the vehicles they produce are designed and fitted with features that help protect people in a crash

- usually you need a force to move something because you're having to overcome another force
- the things being moved has kinetic energy while its moving
- the kinetic energy is transferred to another form of energy when the movement stops

- the word 'work' in physics means the amount of energy transferred from one form to another when a force causes a movement of some sort

- when a car tows a caravan, it applies a force to the caravan to move it to where it's wanted
- to find out how much work has been done, you need to use the equation:
work done (W) (measured in J)  = force (F) x distance (s)

- work is the energy thats been changed from one form to another and is not necessarily the total energy
- the distance needs to be measured in metres
- the force F will be a fixed value in any calculations, either because its constant or because its the average force
- the equation assumes that the direction of the force is the same as the direction of movement
- the equation gives you the definition of the J:
' one joule is the work done when a force of 1 newton moves an object through a distance of 1m'

- sometimes the direction of movement is different from the direction of the force

Example
- to calculate the work done in a situation like a person pulling a sledge with rope attached, you need to consider the horizontal and vertical components of the force

- the only movement is in the horizontal direction
- this means the vertical force is not causing any motion and hence is not doing any work
- the vertical force is just balancing out some of the weight, meaning there's a smaller reaction force

- the horizontal force is causing the motion so to calculate the work done, this is the only force that you need to consider
- this gives us W = Fscos(theta)
- where (theta) is the angle between the direction of the force and the direction of motion

- for a variable force, you cant just use the formula W = Fs
- however, plotting a graph of force against distance moved lets you calculate the work done by just finding the area under the graph
- you might need to split it up into sections that make simple shapes and then add the results together

example
- the graph shows the force exerted by Tibalt the circus monkey as he cycled up a hill
- work done in section A = 40(450/2) = 9000J
- work done in section B = 40 x (350/2) = 7000J
- total work done = 16000J

- power is the rate of doing work 
- the amount of energy transferred from one form to another per second
- power = change in energy (work done) / change in time
- where power is measured in watts
- the watt is defined as a rate of energy transfer equal to one joule per second

- power is also force x velocity
- p = change in W / change in t
- change in W = F x change in s 
- this gives p = (F x change in s) / change in t
- but, v = change in s / change in t which you can substitute into the equation above to give:
- P = Fv

"Energy cannot be created or destroyed. Energy can be transferred from one form to another but the total amount of energy in a closed system will not change"

total energy in = total energy out

efficiency = useful output / total input
multiply by 100 to get efficiency percentage

- the principle of conservation is important with regards to changes between kinetic and potential energy

- kinetic energy is energy of anything moving, which you work out from 1/2mv² 
- there are different types of potential energy e.g. gravitational and elastic
- gravitational potential energy is the energy something gains if you lift it up
~ you work it out using E = mgh
- elastic strain energy (EPE) is the energy you get in a stretched rubber band or spring
~ you work it out using E = 1/2 x k x l² where l is the extension and k is the spring constant

- the energy you need to do things comes from your food
- chemical energy inside the food is converted to other forms e.g. kinetic energy
- 

- you need to be able to use conservation of mechanical energy (change in potential energy = change in kinetic energy) to solve problems such as a simple pendulum
- sin a simple pendulum, you assume that all the mass is in the bob at the end

example
a simple pendulum has a mass of 0.72kg and a length of 0.5m
it is pulled out to an angle of 30 degrees from the vertical
find the gravitational potential energy stored in the pendulum bob

- you can work out the increase in height of the end of the pendulum using trig
- GPE = mgh
= 0.72 x 9.81 x (0.5 - (0.5cos30))
= 0.473...J
= 0.47 J 

the pendulum is released
find the maximum speed of the pendulum as it passes the vertical position
- to find the maximum speed, assume no air resistance, then mgh = 1/2mv² 
- cancel the ms and rearrange to give: v² = 2gh
- v² = 2 x 9.81 x (0.5 - (0.5cos30)
- v² = 1.31429...
- v² = 1.3 ms^-1