Understanding Work, Energy, and Power: Physics Concepts Explained

in this video we're going to talk about work energy and power what is work

what is energy and what is power and how do these three things relate to each other

well let's begin our discussion with work work is something that is accomplished

by the action of a force so let's say you have this block that's resting

on this horizontal surface let's apply a force when this force acts on the block

and moves it by some displacement d that force is going to do work on this

object the work accomplished by the force is the product of the magnitude of the

force times its displacement

now sometimes these two vectors may not always be parallel to each other so let's say if you're pulling the block

with a tension force to the right at some angle but the block is moving

along the x direction so the displacement vector will also be along the x direction

in this case the work accomplished by that tension force is going to be the product of the

tension force times the displacement times cosine of the angle between those

two vectors now let's talk about energy an object with energy

has the ability to do work anytime a force acts on an object what's really

happening is that the force is transferring energy to the object

but before we get into that discussion let's talk about two different forms of energy that

you'll typically encounter in physics the first one is kinetic energy

so what is kinetic energy if you think about the word kinetic the word kinetic

carries the idea of motion so kinetic energy is basically kinetic energy is present whenever you

have an object in motion so let's say if you have a ball that's moving that ball has kinetic

energy potential energy is

basically stored energy so let's say if you have a block that is above ground level that block

has potential energy it has the ability of to fall and when it falls it can do work

so any object that has energy has the ability to do work to calculate kinetic energy

it's equal to one-half mv squared where m is the mass in kilograms

and v is the speed in meters per second when using these units

you're going to get the kinetic energy in joules now potential energy which is a type of

stored energy but this one is particularly gravitational potential energy

since this block has the ability to fall under the influence of gravity

gravitational potential energy is equal to mgh

it's the mass in kilograms times the gravitational acceleration which is in meters per second squared

times the height which is in meters and this will give us the gravitational potential energy in joules

now some textbooks will use the formula u for potential energy and you might see

like a subscript g which indicates gravitational potential energy so you might see the formula like this in your

textbook as for me sometimes i use pe you know this is a habit pe

potential energy i mean it makes sense

so just be aware that you might see these two variables

which correlates to potential energy but sometimes it's better to use this one because there are different types of

potential energy out there you have gravitational potential energy you have elastic potential energy when

you're dealing with springs there's also electric potential energy chemical potential energy so there's a

lot of different types and this might be very useful when you have to distinguish between those uh

different types of energy now let's say if you have a ball

that is moving so because it's in motion it has kinetic energy

and imagine a block being that's placed on this surface and the block is at rest and let's say

there's no friction on this horizontal surface so this ball has kinetic energy

and this block doesn't have any kinetic energy nor does it have potential energy what's going to happen

when the ball strikes the block so let me draw another picture

so when the ball collides with the block what's going to happen in terms of the forces

that are acting on these two now based on newton's third law we know that for every action force

or let me say this again for every action there is an equal and opposite reaction force

when the ball strikes the block it's going to exert a force on a block we can call that the action force

and any time an object exerts a force on another object the first object

is doing work on the second object the first one is exerting the force on the second one

now at the same time the second object is going to do work on the first object

the second object is going to exert an equal but opposite reaction force and any time you have a force acting on

an object there's going to be a transfer of energy each force is going to do work on the

object that it's acting on now the network acting on an object is equal to the change in the kinetic

energy of the object this is basically the work energy theorem whenever a force

increases the kinetic energy of the object that force is doing positive work on the

object it's causing the object to speed up whenever a force

decreases the kinetic energy of the object it's doing a negative work on the object

so in this case the ball object number one it's exerting a force on block 2. it's

causing block 2 to speed up so the ball is going to do positive work on

the object it's transferring some of its kinetic energy to the block

object too now while that's happening block two exerts a reaction force

on ball one as that is happening block two is slowing down ball one

and so block two it's it's exerting it's doing negative work on the ball

because it's slowing down it's decreasing the ball's kinetic energy and so when these two objects collide

there's a transfer of energy the ball is transferring some of its kinetic energy to the block

and so this ball slows down whereas at this block it speeds up another way to determine whether or not

a force does positive or negative work on an object is to consider the direction of the

force and displacement vectors when the force and displacement vectors if they're parallel to each other if

they're pointing in the same direction the work done by the force on the object is positive

now if the force and the displacement vectors are going in the opposite direction so

let's see if they're 180 degrees from each other then the work done is negative

cosine of 180 is negative 1. when they're going in the same direction the angle between them is zero cosine of

zero is one now what about when the force and displacement vectors

when they're perpendicular from each other what can you say about the work done

let's say there's a 90 degree angle well keep in mind the work done by a force is fd cosine beta

so in this case the angle is 90. cosine 90 is zero

so any time the force and the displacement vectors if they're at right angles of each other

then the force does no work on the object it simply causes

the object to turn so the force will become basically

it's going to behave like a centripetal force causing the object to turn into a circle

now let's analyze what we have here during the collision if we focus on the block

we have an action force that is acting on a block and it's pointing to the right

and during the collision the block is going to move to the right as well

so the the force and the displacement vectors they're in the same direction

so the action force is doing positive work on the block

now let's focus on the ball during the collision the ball is moving to the right

so its displacement is in the positive x direction

the reaction force the force that block 2 exerts on ball 1

that reaction force is directed to the left so because these two vectors

are going in opposite directions the work done by the reaction force is negative

so the reaction force is slowing down ball one and the action force is speeding up

block two now consider two situations in the first situation

let's call this ball one ball one is was thrown into the air it's going up

in the second situation we have ball two ball two is falling down under the influence of

gravity now for these two situations is the force of gravity doing positive

work or negative work what would you say well let's focus on the first one

gravity is a downward force gravity likes to bring things down to the ground

so based on this picture the force of gravity will be in the negative y direction

now because the the ball is moving upward the displacement vector

is going to be pointed upward as well now since these two vectors are

going in the opposite direction we could say that the force of gravity is doing negative work

on this ball while it's going up now what about ball two

well gravitational force will always be in the downward direction and this time the ball is going down so

its displacement vector is also in the negative y direction and we know that work

is force times displacement so when you multiply two negative values you're gonna get a positive answer

so the work the work done by gravity whenever an object is falling

that work will be positive now let's think about this in terms of

kinetic energy when the ball is going in the upward direction

the ball is slowing down it's going to go up eventually it's going to be at rest and

then it's going to fall back down so while it's going up the speed

is decreasing if the speed is decreasing what can you say about the kinetic

energy is the kinetic energy increasing or decreasing if the speed is decreasing the kinetic

energy is decreasing the kinetic energy is directly proportional to the square of the speed

so whenever an object speeds up the kinetic energy increases if it slows down the kinetic energy decreases

now if the kinetic energy is decreasing then the change in kinetic energy must be negative

and according to the work energy theorem the network acting on an object is equal

to the change in the kinetic energy of that object so if delta ke is negative

then the work will also be negative so that's how we can determine

the sign of the network on the object so that since

gravity is the only force acting on the object we can say that gravity is doing negative work on this

object now let's focus on ball two is ball two speeding up or slowing down

and how do you know notice that the force and the velocity vectors are in the same direction

when those two are in the same direction the object is going to speed up

in the first example the force and the velocity vectors are in opposite directions so the object

slows down this object is experiencing you could say negative acceleration

it's moving up but the acceleration which is always in the direction of the net force

they're they're opposite to each other when i mean opposite the acceleration is opposite to the velocity vector but the

acceleration is always parallel to the net force so since the acceleration is in the

negative y direction and the velocity is in the positive y direction the object is slowing down

it's experiencing deceleration but for for ball two it's speeding up the force and therefore the acceleration

is in the same direction as the velocity vector so if it's speeding up

we know that the kinetic energy of ball two is increasing therefore the change

in kinetic energy for ball two will be positive if the change is positive then the work

done on ball two is positive so we could say that gravity is doing positive work on ball two it's speeding it's beating

the ball up and for ball one gravity does negative work on it because gravity is slowing it

down it's decreasing its kinetic energy now what about potential energy gravitational potential energy

depends on the object's position relative to some reference point so

basically depends on the height of the object as well as its mass now ball one is moving up

it's moving away from the ground as it does so the height between

ball one and the ground increases therefore the potential energy of ball one is increasing as it moves

away from the ground ball two is moving towards the ground so the height difference between the

ground and the position of ball two is decreasing so the potential energy of ball two is

decreasing now these two are usually not always but usually inversely related when one goes

up the other goes down in the case of ball one as ball one goes up

its kinetic energy is being converted to potential energy the kinetic energy is decreasing the potential energy is

increasing in the case of ball 2 the potential energy is being converted

into kinetic energy as it falls it's losing potential energy but it's speeding up it's gaining

kinetic energy now the sum total of an object's kinetic and potential

energy is the mechanical energy energy is conserved

when the only forces acting on the object are conservative forces

gravity is a conservative force here gravity decreases the kinetic energy of the

object but here it increases it now gravity is not the only type of conservative force that you need to be

familiar with there are other types of conservative forces

these include the elastic force associated with springs and the electric force

interestingly all these three types of forces have their own types of potential energy

like gravitational potential energy elastic potential energy or electric potential energy

now non-conservative forces they do not conserve the mechanical energy of an

object friction is a non-conservative force friction will always slow down an object

it will never speed it up air resistance is another type of non-conservative

force and then push and pull actions

those type of forces let's say if you're trying to push an object that doesn't conserve the mechanical

energy and that can increase or decrease the mechanical energy another one is the tension force that's

a non-conservative action force for instance let's say if you have an object

that is above ground level and you apply an action force

that is greater than the gravitational force that's pulling it down

let's say the action force that you're applying is significantly greater than the gravitational force

that means that there's going to be a net force pushing this object up now notice what's happening here

because we have a net force there is a net acceleration in the y

direction which means the object is speeding up in the y direction

so it's moving upward and because there's an acceleration

it's speeding up therefore the kinetic energy is increasing now because you're moving it away from

ground you're increasing the height of the object at the same time therefore the potential energy is increasing

so in this in this scenario you're increasing both the kinetic and the potential energy

in the last scenario one of these went up the other went down in this scenario both of these things

both of these forms of energies are going up now mechanical energy which is the sum

of these two kinetic and potential is also increasing because if the kinetic end of potential

energy is increasing then mechanical energy is increasing so in this case the mechanical energy is

not conserved you're using an action force to not only speed up the object to increase its kinetic energy but you're

also increasing its stored gravitational potential energy as you move it away from ground level

so you're increasing the object's mechanical energy therefore this action force is a

non-conservative force mechanical energy is not conserved when an action force

is acting upon an object so the work done by this action force

is positive because the object's kinetic energy is

increasing also this force

is in the same direction as the object's displacement so that force is doing positive work on

the object now this object has multiple forces acting on it and each force does its own

type of work on its object the action force is clearly doing positive work on the object

gravity is doing negative work on the object as you can see

the force of gravity and the displacement vector are in opposite directions

now what about the network on the object is it positive or negative well the net force

is in the positive y direction and it's in the same direction as the displacement vector so the net work done

on this object is positive it's based on the direction of these two vectors

now let's talk about power what is power power is related to work

but power is a rate it's the rate at which work is done on an object

it's also the rate at which energy is transferred from one object to another power

which will use the symbol p power is equal to work divided by time

so an object that can do work in a short amount of time is exerting a lot of power

so power is the rate at which energy flows work is typically in units of joules

power is i mean time is usually in seconds and power is usually in watts

one watt is equal to one joule per second a kilowatt

is equal to a thousand watts a megawatt

is basically a million watts one times 10 to the six watts and a horsepower

is 746 watts so those are some units that you want to

be familiar with so remember power is work over time and it's the rate at which energy is being

transferred another equation for power is force

times velocity if you know the force acting on an object

and you know the object's velocity you can also calculate the power and we could derive that equation from

this one work we know that work is force times

displacement and time is just t now displacement over time

that's equal to velocity so we can replace d over t with v so we get power is force times velocity

so that's another way in which you can calculate the power being exerted

now let's use a another way to illustrate the concept of power so let's say we have two individuals

we'll call this person john and this one jared

let's say that john lifts a 100 newton box

a distance of one meter

above the ground so work is force times displacement so a force of 100 newtons times a

displacement of one meter he's doing a hundred joules of work now let's say jared does the same amount

of work he also lifts a 100 newton box

one meter above the ground so he's doing a hundred joules of work but now let's say that john

he takes one second to lift up that box

but let's say jared it takes him 10 seconds to get the job done which individual exerts more power would

you say it's john or jared well we know that power is work over time

it took john one second to get the job done so the amount of power that he exerted

is a hundred watts it's a hundred joules per one second so he's exerting the power of 100 joules

every second now jared it took him 10 seconds to get the job done

so if we take his work divided by his time it's 100 joules per 10 seconds or 10

watts so john he transfers 100 joules in one second

jared he's only transferring 10 jewels per one second so we could say that john is more

powerful it took him a short amount of time to get the same job done whereas jared it

takes him a longer amount of time to get the job done in one second john can transfer a hundred joules of

energy in one second jared can only transfer 10 joules of

energy so in this way you could see two different ways in which you can view

power power the more power you have

the faster you can get the job done the less power you have the longer it's going to take you

to get the same amount of work done so you can think of power as work over time or the rate at which energy is

transferred john has a greater rate at transfer energy in

one second he can transfer 100 joules of energy and jared

his rate of energy transfer is much less in one second he can only transfer 10 joules of energy

so hopefully this illustration helps you to understand the concept of power as being the rate at which energy is

transferred now let's work on some practice problems number one

what is the kinetic energy of a five kilogram block sliding across a frictionless horizontal

surface at 12 meters per second so let's begin with a picture

okay that not that line is not very horizontal let's draw a better one

so here is our five kilogram block and it's moving horizontally at a speed of 12 meters per

second to calculate the kinetic energy we can use this formula

ke is equal to one-half mv squared so the mass of the block is five

kilograms and the speed is 12 meters per second 12 squared or 12 times 12 that's 144

half of 144 is 72. so it's 72 times 5 7 times 5 is 35

so 70 times 5 is 350 and 2 times 5 is 10 so when you add 350 and 10 you get 360. so this block has 360

joules of kinetic energy and that's the answer for part a number two

what happens to an object's kinetic energy if the mass is doubled

let's start with the equation ke is equal to one-half mv squared

for these types of problems we need to do is replace everything that doesn't change

or that remains constant with a one and then what changes plug in the appropriate number

so one half i mean that's a constant we're just going to replace with a one the mass it doubles we're

gonna replace it with a two the speed doesn't change so we're gonna plug in a one

and this will give us two what this tells us is that if you double the mass the object's kinetic energy will double

now if we move on to part b what's going to happen to the speed i mean what's going to happen to the kinetic energy if

we double the speed so one half doesn't change we're gonna replace it with a one

the mass doesn't change but this time we're doubling the speed two squared is four

so if you double the speed the object's kinetic energy will increase by a factor of four

now what if you triple the speed three squared is nine the object's kinetic energy will

increase by a factor of nine now what about part d what if we triple the mass

and we quadruple the speed so we're going to replace m with 3 v with 4.

4 squared is 16. three times sixteen is forty eight so in this case the object's kinetic

energy will increase by a factor of forty eight number three

what is the gravitational potential energy of a 2.5 kilogram book that is 10 meters above the ground

so let's draw a picture so here is the 2.5 kilogram book

and it's 10 meters above ground level so it has the ability to fall how can we calculate the gravitational

potential energy the gravitational potential energy is going to be equal to mgh

so the mass is 2.5 kilograms the gravitational acceleration is 9.8 meters per second squared

and the height is 10 meters 2.5 times 9.8 times 10.

so this is equal to 245 joules

so that's the gravitational potential energy of this book

when it's 10 meters above the ground a 10 kilogram ball falls from a height of a hundred meters

calculate the vertical speed of the ball during the first four seconds so let's say this is the ground level

and here's a ball and it falls down and it's a hundred meters

above ground level how can we calculate the vertical speed of the ball during the first four

seconds so i'm going to make a table i'm going to put time

vertical speed with v y after that we need to calculate the height

of the ball and then the kinetic energy potential energy

and finally the mechanical energy so at t equals zero the vertical speed of the ball is zero

because it's released from rest now based on this equation v final is equal

to v initial plus a t now the initial speed we said it's zero so the final speed in the y direction

is the acceleration which the acceleration in the y direction

is gravitational acceleration times t so every second the vertical speed is going to increase by 9.8 meters per

second so at t equals one it's going to be 9.8 times 1 or simply 9.8 at t equals 2

it's going to be 9.8 times two so it's nineteen point six meters per second

and at t equals three it's nine point eight time stream so that's

29.4 meters per second and 4 times 9.8

is 39.2 so as you can see each second the vertical speed increases by 9.8

which is the gravitational acceleration acceleration tells you how much the speed is going to change

every second now what about the height so at t equals 0

the ball is 100 meters above the ground what is it going to be one second later what equation can help us

to find the distance that the ball travels we could use this equation displacement

is equal to v initial t plus one half a t squared now the initial speed in the y direction

is zero so this term is zero so therefore the displacement in the y

direction is one half g t squared because acceleration in the y direction is g

so it's going to be 0.5 times negative 9.8 times

a time of just one second so 1 squared that's negative 4.9 so what

that means is that the ball fell down a distance of 4.9 meters

so if it goes down 4.9 meters 100 minus 4.9 is 95.1 it means that it's 95.1 meters

above the ground so two seconds later that's gonna be 0.5 you can use positive 9.8 if you want as

long as you understand what's happening 0.5 times 9.8 times 2 squared that's 19.6

so it fell down by 19.6 if you use negative 9.8 it will be negative 19.6 the negative sign simply tells you

that it falls down by 19.6 meters now if you're looking for the velocity as opposed to speed

these will all be negative values so when you use the equation v final is equal to gt

g is negative so this will give you velocity but we don't really need velocity

in this problem because kinetic energy is based on speed so that's why i chose to use positive

values but if you're dealing with velocity because the ball is going in the

negative y direction these values should have negative velocity values

however since speed is positive we don't have to worry about that so now let's calculate the height two

seconds later so if the displacement is negative 19.6 we need to subtract 19.6 from 100.

so the height two seconds later is now 80.4 meters what about three seconds later

so d y is going to be one half negative nine point eight times three squared

so that's negative four point nine times three squared and so that's the displacement of

negative 44.1 so let's subtract that from 100 and so we're going to have a height

of 55.9 meters now let's do the same thing for

a time of 4 seconds so negative 4.9 times 4 squared is negative

78.4 meters so 100 minus 78.4

is 21.6 meters so that's the height above the ground four seconds later

now let's calculate the kinetic energy of this ball for each second

now at t equals zero because the speed is zero the kinetic energy will be zero one second later it's going to be one

half times a mass of 10

times v squared which is 9.8 squared so half of 10 is 5 so 5 times 9.8 squared

that's going to give us an energy of 480.2 now let's repeat the process so the next

one is going to be one half times 10 which

is basically going to be 5. the mass is not going to change so five times nineteen point six squared

so that's one thousand nine hundred twenty point eight and then five times twenty nine 29.4

squared that's 4321.8 and then 5 times 39.2 squared

that's 7600 now let's move on to the next column let's calculate the potential energy

at the different times so potential energy is mgh so it's based on the height so the mass

is always going to be 10 g is 9.8 and the height is the stuff that's going to change

so initially the height is 100. so it's going to be 98 times the height so 98 times 100

is 9800 joules of energy

now what about when the height is 95.1 so the 98 part is going to stay the same and then the height is just 95.1

so 98 times 95.1 is hundred 9319

point eight next is going to be ninety eight times eighty point four

and so that's seven thousand eight hundred seventy nine point i'm dealing with some very small spaces

i should have made this larger so next is 98 times 55.9 which is 5478

and then finally 98 times 21.6 which is 2116.8

so now what i'm going to do is calculate the total mechanical energy the mechanical energy of a system

is the sum of the kinetic energies and the potential energies so for the first one it's 0 plus 9 800

so that's going to be 9 800. next if we add

these two values so 480.2 plus 93 19.8 you get the same answer

9 800. and then if we add those two values

so 1920.8 plus 78 79.2 you still get

9 800. and then 43 21.8 plus 5478.2 and i think

you see the picture now just make sure everything is correct i'm going to add the last one as well

make sure i miss anything so what does this all mean what can we learn from this

so we see that the total mechanical energy is conserved it's a constant value

doesn't change it remains 9 800 joules so what's happening is that as the ball

falls the potential energy is being converted to kinetic energy notice that the potential energy

decreases from 9800 to 21 16.8

once it hits the ground the potential on julie zero and while it's falling the kinetic

energy increases the object is falling down with greater speed the speed is increasing

and so as the potential energy decreases the kinetic energy increases allowing the mechanical energy to remain

constant now what about the last part is gravity a conservative force

it turns out it is the only force acting on the ball is the gravitational force

whenever you have a system in which only conservative forces are acting on the system

the total mechanical energy will be conserved if you have a non-conservative force

like friction friction will decrease the mechanical energy of the system eventually

the object will come to a stop if you roll a ball on a carpet the ball will eventually come to a stop

because friction is going to slow it down to a stop and so friction reduces the mechanical

energy of the system now let's say if you apply a force to accelerate the object you can

increase its overall mechanical energy so an applied force or frictional force these are both non-conservative forces

they can increase or decrease the mechanical energy of the system but gravity is an it's a conservative force

it doesn't change the mechanical energy of a system so that's why

the mechanical energy is constant because we only have a conservative force acting on the ball

so this problem says that a 70 newton force is applied horizontally

to a 10 kilogram block at rest for a displacement of 200 meters across a frictionless surface

so this is going to be the horizontal frictional surface and this

is the 10 kilogram block so we're going to apply a force of 70 newtons

now the block is going to travel for a displacement of

200 meters so the force is going to be applied

until the block travels the distance of 200 meters during that time we want to calculate

how much work is done by the force as it travels from point a to point b

so the work done by the force is the force times the displacement so the force is 70 newtons

and the displacement is 200 meters so it's 70 times 200 7 times 2 is 14 so we just got to add

the four zeros i mean the three zeros so the work done by the force is 14 000 joules

now this force is the only force acting on the block in the x direction so this force represents

the net force as well so we can say that this is also the network

done on the block because there's no other forces acting on it in the x direction

so this is the answer to part a now we could set the network equal to the change in kinetic energy so

that is the final kinetic energy minus the initial kinetic energy

now at point a the block is at rest which means it's not moving the speed of

the block is zero and if the speed is zero the kinetic energy is zero

so the initial kinetic energy is zero which means the network is equal to the final

kinetic energy which is 14 000 joules so part a and part b is the same it's

both 14 000 joules so now we can move on to part c the final kinetic energy is equal to

one-half mv squared and the final kinetic energy is 14 000

and the mass is 10 so we can calculate the final speed half of 10 is five

and so i'm going to take 14 000 and divide it by five and so that's equal to 2800 so 2800 is

equal to the square of the final speed so if we take the square root of both

sides this will give us the final speed so the final speed at part b

i mean a part b but point b rather which is the square root of 2800 that's 52.9

meters per second so that's the answer to part c now let's move on to part d

what is the acceleration of the block in the horizontal direction well we know that the net force

is equal to this force because that's the only force

in the x direction and the net force in the x direction is the mass times the acceleration

in the x direction according to newton's second law so f equals m a

now we have a mass of 10 and the force applied is 70 newtons so the acceleration is going to be 70

divided by 10 which is 7 meters per second squared

so now that we have the acceleration let's move on to part e use kinematics to calculate the final

speed of the block let's confirm the answer that we have what kinematic formula do you know can

help us to calculate that final speed so we have the acceleration we know the initial speed is zero and we have the

displacement so the equation that has the initial speed final speed acceleration and

displacement is this equation v final squared

is equal to v initial squared plus 2ad the initial speed is zero the

acceleration is seven and the displacement is two hundred

so uh two times seven is fourteen and fourteen times 200 well 40 times 2 is 28 so 14 times 200 is

2800 so we can see that this is going to lead us to the same answer

if we take the square root of 2800 that's going to be 52.9 meters per second

so as you can see there's multiple ways in which you can find the final speed of the block you could use kinematics

or you could use the work energy principle in fact if you combine the two methods

you can derive the equation for the kinetic energy of an object

so we know that the network is equal to the change in kinetic energy

but to prove that the network also equals to the force times the displacement

and force is mass times acceleration now using this equation v final squared

is equal to v initial squared plus 2ad so i'm going to subtract both sides by v initial squared

so i have v final squared minus v initial squared is equal to 2ad

next multiply both sides by a half so on the left side i'm going to have

one half v final squared minus one half

of the initial squared and one half of 2ad is simply a times d because one half times two is one

so what i'm going to do now is i'm going to replace a d with this expression

because they equal each other so the net work done on an object is going to be the mass

times one half v final squared minus one half of the initial

squared so if we distribute the mass it's going to be one half

mv final squared minus one half mv initial square

and so this expression is the final kinetic energy and this expression

is the initial kinetic energy so therefore we can say that kinetic

energy is one-half mv squared

and that's how you could derive the formula for kinetic energy as you can see here

and so the final kinetic energy minus the initial kinetic energy is equal to the change

in kinetic energy so we have this principle the work energy principle that is the net work done on an object

is equal to the change in the kinetic energy of that object so keep this principle in mind it's very

useful how much work is required to accelerate a 1500 kilogram car

from 15 meters per second to 40 meters per second so keep in mind the net work

is equal to the change in the kinetic energy so that's the final kinetic energy

minus the initial kinetic energy so it's one half mv final squared

minus one half mv initial squared

now to make the calculation a lot easier we can factor out one half m because it's the gcf

so the net work is also equal to one half m times the v final squared

minus the initial squared so the mass of the car is 1500 kilograms

the final speed is 40 and the initial speed is 15. so half of 1500

is 750 and 40 squared is 1600 15 squared is 225

and 1600 minus 225 is 13.75 so let's multiply 1375 by 750

so you should get 1 million 31 and 250 joules

so that's the network done on the car now let's move on to part b what is the average net force acting on

the car if it reaches a final speed of 40 meters per second

while traveling a distance of 275 meters so we know that the network is equal to the net force

times the displacement distance and displacement is the same if you have an object traveling in one

direction if it doesn't change direction so we have the network it's about a million joules

our goal is to calculate the average net force and the displacement is 275 meters

so all we got to do is take the work that we have and divided by 275

so you should get an average net force of 3750 newtons

so let's see if we can get this answer using another technique

the second way in which we could calculate the average net force is using kinematics we need to find the

acceleration first so let's use this formula to calculate the acceleration

the final speed is 40 the initial speed is 15 and the displacement is 275

so we know 40 squared is 1600 15 squared is 225 and 2 times 275 is 550.

now 1600 minus 225 is 13.75

so to calculate the acceleration we need to divide both sides by 550. 1375 divided by 550

is 2.5 meters per second squared so that's the acceleration of the vehicle

so now we can calculate the net force using newton's secular

net force is equal to ma so the mass of the vehicle is 1500 kilograms

multiplied by an acceleration of 2.5 meters per second squared so 1500 times 2.5

as we expect is 3750 newtons so the answer is confirmed

how much work is done by a constant 50 newton force that acts over displacement of 10 meters

to find the work done by a constant force it's simply the force times the displacement

assuming that they're both parallel to each other so it's going to be 50 newtons

times the displacement of 10 which is 500 joules now what about part b

how much work is done by a varying force that increases at a constant rate from 40 to 80 over displacement of 10.

so if you have a force that's not constant but it's changing at a constant rate

then the work done by that force is equal to the average force times the displacement

the average force is basically the initial force plus the final force divided by two

multiplied by the displacement so it's going to be one half times the sum

of the initial and the final force times the displacement of 10. 40 plus 80 is 120 and half of 120 is 60.

so the average force is 60 which is between 40 and 80. so it's 60 times 10

which means it's 600 joules now you can confirm these answers graphically

in the case of the first example let's make a graph so i'm going to put the force vector on the y axis and

displacement on the x-axis now the force is a constant 50. so let's put 50 here

and the displacement is 10. so if you have a force displacement graph

the work done by this force is equal to the area under the curve

and so we have is a rectangle the area of a rectangle is length times width

so the length in this example is the force the width is the displacement so it's 50 times 10

and so the work done is 100 or rather 500 joules

now for the second example we can graph it as well so this is going to be the force and the

displacement vector now the force increases from 40 to 80

over a displacement of 10 meters so at zero the force is 40 and at 10 is 80.

so it increases at a constant rate so what we need to do is find the area of the shader region

how can we do so for this type of graph you want to split it into two parts

into a rectangle and a triangle the area of the bottom rectangle is the length times the width so it's 40 times

10 which is 400 joules now we need to calculate the area of the

triangle which is uh one-half base times height

the base of the triangle is 10 meters and the height of the triangle

is the difference between 80 and 40 which is 40. now 10 times 40 is 400 and half of that

is two hundred so the total area under the curve

is two hundred plus four hundred which gives you six hundred so that's how you can calculate the work

done by a varying force you can use this equation if the force changes at a constant rate

or you could simply find the area under the curve you