in this article we're going to talk about kinematics which basically describes how objects move without any references to force now we're going to focus on kinematics in one dimension mostly along the x-axis but we can also work on some problems along the y-axis when you get into two-dimensional kinematics it covers projectile motion which is a topic for another day now the first thing we need to talk about is the difference between a scalar quantity and a vector quantity a scalar quantity is something that has magnitude only whereas the vector quantity has both magnitude and direction so for instance mass is a scalar quantity you could say that a book has two kilograms of mass the two kilograms would be the magnitude of the mass of the book but you wouldn't say the book has two kilograms of mass east direction wouldn't apply for mass so mass cannot be a vector quantity distance is a scalar quantity and displacement is a vector quantity displacement you can think of it as distance with direction so let's say if a person traveled 15 meters you're basically describing the person's distance because you didn't mention direction but let's say if someone traveled 15 meters east now you're describing displacement you describe in not only how far he traveled but also where he traveled so think of displacement as distance with direction now technically displacement is really the change in the position of an object so along the x-axis you could describe displacement as the final position minus the initial position speed is a scalar quantity speed describes how fast an object is moving velocity is a vector quantity velocity is speed with direction so if you were to say a person is traveling at uh 20 meters per second you're describing the speed of the person this is the magnitude but let's say if a person is moving at 30 meters per second north you're describing velocity because you've mentioned his speed along with the direction so you have both magnitude and direction which makes it a vector quantity now what about temperature would you say that temperature is a scalar quantity or would you describe it as a vector quantity so think of units of temperature let's say it's uh 80 degrees fahrenheit outside or 25 degrees celsius you wouldn't say it's 80 degrees fahrenheit north doesn't apply for temperature so because you can't describe the direction of temperature it's not relevant temperature is a scalar quantity you can only mention its magnitude such as how hot it is 80 degrees fahrenheit but you can attach direction to temperature which makes it scalar now acceleration that is a vector quantity acceleration tells you how fast the velocity is changing and you could say a car is accelerating at 2 meters per second squared east you can put direction to it so acceleration is the vector quantity now let's talk more about the difference between distance and displacement so let's say a person walk let's say a person traveled 30 meters east and then he turns around and travels four meters west we're going to add a negative sign to this because he's going towards the left and just to review direction north and south is along the y-axis east and west they're along the x-axis so east is along the positive x-axis west is along the negative x-axis so given a situation calculate the distance and the displacement of this individual that uh traveled here let's put a person so the total distance that this person traveled is 17 meters it's basically the sum of these two numbers he traveled 30 meters east and then 4 meters west so a total distance of 17 meters distance being scalar it's always positive now there is an exception to this specifically temperature depending on the units you're dealing with even though temperature is a scalar quantity you could have negative values especially for units such as fahrenheit and celsius so some places can be negative 20 degrees fahrenheit in temperature others negative 10 degrees celsius but on the kelvin scale there's no negative values so there are some scalar quantities that do have negative values like temperature so just be aware of that now going back to this problem now that we know the total distance traveled what is the displacement of this particular individual now displacement can be positive or negative dependent on the direction if you were to combine these two including the negative 4 value it would be 13 minus 4 so the displacement is positive 9. displacement is the final position minus the initial position so displacement is the change in position what this tells us is that this person ended up nine meters east from where he started and so let's illustrate this with a number line so let's say that he started at the origin position zero he traveled 13 meters east so because he's moving along the positive x-axis we assign a positive value here and then during the second part of his trip he traveled 4 meters west so 13-4 he ends up at position nine so the net result is that he trave led nine meters east and so that's his displacement that's how far he traveled uh relative to his initial his initial position so hopefully this example gave you a good understanding between the difference of the distance and displacement so remember distance is a scalar quantity it's always positive and displacement is a vector it has both magnitude and direction now let's talk about the difference between speed and velocity remember speed is a scalar quantity but velocity is a vector quantity speed has magnitude only but velocity has both magnitude and direction let's use s to describe speed s would represent the instantaneous speed that is how fast an object is moving at an instant of time but s bar represents the average speed average speed is equal to the distance traveled divided by the time that was elapsed average velocity is equal to displacement over time so therefore speed is always positive velocity can be positive or negative dependent on the direction so average velocity is basically the velocity calculated over a time interval now let's use an example to calculate speed i mean average speed and average velocity so let's say we have a particle and this particle travels 100 meters east and then it turns around and travels 150 meters west and it does all of this in five seconds calculate the average speed and the average velocity of this particle over this 5 second time interval so to calculate the average speed first we need to calculate the total distance traveled this particle traveled a total distance of 250 meters it traveled 100 meters east and then 150 meters west so if you add those two numbers you get a total distance of 250 meters now it did this in five seconds 250 divided by five is 50. so the average speed of this particle is positive 50 meters per second now the average velocity is different because the displacement is different now there's two ways in which we can calculate the displacement we can add up the individual the displacements for the individual segments of the problem the displacement for the first part is 100 meters and the displacement for the second part is negative 150. if we add a 100 plus negative 150 we get a net displacement of negative 50. the other way in which you can calculate displacement is by taking the final position and subtracting it by the initial position and it helps to draw a number line for this so the particle started at position zero and during the first part of his traveled it was at position 100 and then it traveled 150 meters west so it ended at position negative 50. so its final position was negative 50 minus its initial position of zero which would still be negative 50. i mean you could do it that way too but i prefer to simply add the displacement of each individual part of the problem to get the final displacement so the final displacement is negative 50 meters divided by a time of 5 seconds so negative 50 divided by 5 the average velocity calculated is negative 10 meters per second so as you can see the average speed and the average velocity is not always the same it's going to be the same if the person or the object travels in one direction but if there's any change in direction like when this particle you know decided to go west and that's when the average speed and the average velocity will be different so keep in mind the average speed doesn't have to equal the average velocity sometimes they're equal to each other but it's not always the case now for instantaneous speed that is the absolute value of instantaneous velocity the instantaneous velocity tells you the velocity at an instant of time whereas the average velocity basically tells you the average over an interval so remember average velocity is the displacement over time and displacement is the final position minus the initial position divided by t now instantaneous velocity in order to calculate it you need to use limits so it's the limit as the change in time goes to zero and it's basically the displacement over time or the change in position over the change in time so that's how you would calculate instantaneous velocity which we really won't go over that in this uh video but for those of you who want the formula that's what it is and whenever you see delta this triangle it represents the change of something in this case the change in position so delta x is the final position minus the initial position so basically it's the displacement along the x-axis but you can also have displacement along the y-axis in this case it will be the final position along the y-axis minus the initial position so whenever you see the symbol d you could describe it using distance or displacement now let's talk about some formulas that you need to be familiar with when an object is moving with constant speed typically this is the formula that you need to work with d is equal to vt so in this equation d can represent distance or displacement depending on how you use the problem v you could use spe ed or velocity it's going to work if you're dealing with constant speed but understand this though if you're using vs speed then d is going to be the distance if you're using v with reference to velocity d is going to represent the displacement now for objects moving at constant speed you need to know that the instantaneous velocity is the same as the average velocity because the velocity is not changing so whether you use v or v bar it'll have the same effect now let's talk about when objects are moving with constant acceleration and by the way remember if you're using displacement displacement is the final position minus the initial position but if you're dealing with distance you don't have to worry about that so just keep in mind if you're using d as displacement or distance now for constant acceleration we have some formulas that we need to take into account for constant acceleration d is equal to v bar times t and the reason for that is that v doesn't equal v bar when there's acceleration any time there's acceleration that means the velocity is changing acceleration tells you how fast the velocity is changing and if the velocity is changing then the instantaneous velocity and the average velocity for most of the time won't be the same so that's we need to use v bar now average velocity is the average of the initial velocity and the final velocity so it's basically the sum of the initial and the final velocity divided by two or you can divide it this way one half the initial plus v final so if we were to replace v bar with this expression we would get the displacement is equal to one half v initial plus v final times t so remember if you're using d as distance then v initial is the initial speed v final is the final speed but if you're using ds displacement v initial is your initial velocity v final is your final velocity so let's rewrite that equation here this is one of those equations that you want to write in your list of equations if you have a test coming up now there's some other equations that we're going to add to this list so we said that average velocity is the displacement which is final position minus initial position divided by t well if you were to rearrange that equation you'll get this one final position is equal to initial position plus the average velocity times t you can also get it from this equation d is equal to v bar times t and uh d being the displacement is x final minus x initial and so if you were to move this to that side you would end up with the same equation so there's many ways in which you can derive that equation so that's the next one that you want to have in your list now we said that average velocity is the displacement or the change of position divided by the time average acceleration is the change in velocity divided by time it's the final velocity minus the initial velocity divided by t so if you rearrange that equation you get something similar to the one that we have above v final is equal to v initial plus a t now there are some other equations that we can add to the list another one is v final squared is equal to v initial squared plus two times a d and then there's this one displacement is equal to v initial t plus one half a t squared and anytime you see the letter d you can replace that with x final minus x initial so if we were to substitute x final minus x initial for d and then move x initial to the right side of the equation we'll get this equation final position is equal to initial position plus v initial t plus one half a t squared now because this is x we're dealing with motion along the x axis but when you go into projectile motion you can apply motion along the y axis as well so you might see this variation of this formula y final equals y initial plus v y initial t so that's initial velocity but in the y direction plus one half a t squared where a is like the gravitational acceleration in the y direction so just understand that you can apply uh these formulas in the x direction or in the y direction so it might be a lot to keep track of but as you begin to work problems uh the use of these formulas will begin to make more sense but you may want to write these down for your reference now before we work on a few example problems there's something i do want to mention when dealing with constant speed you could use this formula x final is equal to x initial plus v t so you don't need to write v bar because v and v bar are the same uh when dealing with constant speed but for constant acceleration it's good to keep in mind that this represents average velocity so it's better to write v bar instead of v so there's no confusion because these two they're not necessarily the same when dealing with constant acceleration so just be aware of that little detail let's start with this one a bus is traveling at a constant speed of 40 meters per second how many hours will it take to travel a distance of 200 miles so what formula do we need to use so we're given a speed which we can use as a symbol v or s and we know the distance which is 200 miles the only formula that we could use since the bus is moving at constant speed is this one d is equal to vt now before we use that formula we need to take a look at the units here we have meters per second and here we have mouse and we want to find a time in hours so we're looking for t the units they don't match so before we can use the formula we need to convert the units into an appropriate form so what do you recommend that we need to do we have the distance in mouse and we want to find a time in hours the best thing we can do is convert the speed from meters per second to miles per hour once we do that then it will match with the unit hours and the unit miles so we can now use that formula so let's go ahead and convert 40 meters per second to miles per hour so let's get an overview of how we're going to do this let's convert meters to kilometers and then kilometers to miles and then we're going to convert seconds to minutes and then minutes to hours so in order to convert miles to kilometers you need to know the relationship between the two one kilometer is equivalent to a thousand meters I meant to say meters instead of miles but that's how you can go from meters to kilometers now these units cancel now let's convert from kilometers to miles so we need to put the unit kilometers on the bottom miles on top so that these units will cancel you need to know that one mile is equal to 1.609 kilometers so now we have the unit mouse let's convert seconds to minutes since we have seconds on the bottom we want to put seconds on top and then minutes on the bottom one minute is equal to 60 seconds so now we can cross out this unit and now let's convert minutes to hours one hour is equivalent to 60 minutes and so we could cancel the unit minutes so now let's do the math we're going to multiply by all the numbers that are on the top but we're going to divide by the numbers on the bottom so it's going to be 40 divided by a thousand take that result divided by 1.609 and then multiply that by 60 and then by another 60. so for the velocity or rather the speed you should get 89.5 if you round it miles per hour which we can write mph so that's how we can convert meters per second to miles per hour now let's go ahead and finish this problem so let's use the formula d is equal to vt in this case d is going to represent the distance which is 200 miles v is going to be the speed which is 89.5 miles per hour and then now we can calculate t so to get t by itself we need to divide both sides by 89.5 miles per hour 200 divided by 89.5 gives us this answer 2.23 hours now looking at the units we can see that the unit miles cancel leaving behind the unit hours which is what we want the final answer to be in so when dealing with these problems always check to make sure that the units match if they don't match you need to convert one unit into another so just keep that in mind now let's move on to part b of this problem if the bus moved from a position that is 50 miles east of city xyz to a position that is 90 miles west of city xyz in five hours what is the average velocity of the bus during that time interval so let's say this is city xyz so the bus was initially let's say at position a which is 50 miles east of city xyz actually that's wes i'll take that back let's say city xyz is at the origin position zero so the bus was 50 miles east of the city so it was initially here so this is going to be x initial that's the initial position of the bus and then it moved to a position that is 90 miles west of that city so this is going to be negative 90. that's the final position of the bus so the bus is traveling west which means that the average velocity should be a negative value now how can we calculate the average velocity what formula do we need to use it really helps to write down everything we know that x initial is 50. x final is negative 90 and the units are in miles and then we have the time which is 5 hours we want to calculate the average velocity well we know that average velocity is displacement over time and displacement is the change in position x final minus x initial whenever you're dealing with motion along the x axis and then divided by the time so the final position is negative 90 the initial position is positive 50. the time is 5 hours so negative 90 minus positive 50 that is negative 140. so the change in position or displacement that is a negative 140 miles and we're going to divide that by five hours so 140 divided by 5 is 28 thus the average velocity is going to be negative 28 miles per hour which we can write as mph so that's how we can calculate the average velocity