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