High School Physics: Vectors and Scalars

Transcript

hi everyone I'm Dan Fullerton and today I'd like to talk to you about vectors and scalers our goals are going to be to learn to differentiate between vector and scalar quantities to use scaled diagrams to represent and manipulate vectors we want to be able to determine X and Y components of vectors and also go the other direction and given X and Y components of vectors be able to find the angle of a vector so to begin with let's talk about scalers scalers are physical quantities that have a magnitude or a size only things like temperature doesn't have a direction it only has a size mass how much stuff you're made up of in kilograms doesn't have a direction associated with it or time which we measure in seconds and you may think of time going forward and backward but that's not really a direction like a direction on the compass so those are all scalar quantities compared to those we also have vectors these are quantities that have a magnitude and a direction things like velocity I'm going 35 m/s due south things like force I'm pushing on something with a force of 50 Newtons and I'm pushing in a specific direction or momentum that truck coming at me barreling at me down the highway has a tremendous momentum and the direction of that momentum is right toward me now we typically represent vectors in physics is arrows and the direction of the arrow of course tells you the direction of the vector and the longer the arrow the bigger the vector so the light blue Vector here let's call that Vector a has about half the magnitude of vector B Vector B has the same direction as Vector a it's twice as long therefore it has twice the magnitude if we assume for example that Vector a represents a force of 10 new Newtons and I go and measure that with my ruler I find that it's actually 4 cm long from tip to tail therefore this must be telling me that there are 10 Newtons over 4 cm or every cmet is roughly 2.5 Newtons so if I then go and I measure Vector B with my ruler I find that it is 8 cm in length so what force would Vector B represent well 8 cm time 2.5 Newtons per cimer tells me that Vector B has a force of 20 newtons which makes sense we said it was twice as big as Vector a if Vector a is 10 Newtons Vector B should be twice as big 20 newtons so we can use vectors in scale diagrams to help us figure out exactly what magnitude of of a quantity we're talking about now adding vectors is a fairly straightforward process graphically here we have two vectors A yellow vector and a light blue Vector if we want to add them what's really nice about vectors is you're allowed to move them around you can't change their direction or their length but you can slide them wherever you want so the trick to adding vectors is to always line up all your vectors so that they are tip to tail so if we slide these around and make these tip to tail our light blue Vector now moves so that its tail is touching the tip of the yellow Vector now our next step is if we draw a line from the starting point of our first Vector to the ending point of our last Vector we get our answer the sum of a vectors A and B is this Vector in Black a plus b which we could determine graphically and it doesn't really matter in what order you place place the vectors we could do the yellow one then the blue one or the blue one and the yellow one for example if we went with the blue one and the yellow one you can see we still get the exact same answer we just went in the opposite order and that will work for any number of vectors from two vectors up to 200 vectors and further on line up all your vectors tip the tail draw a line from the starting point of the first to the ending point of the last and you will end up with the sum of your vectors vector addition and you call the sum of any vectors the resultant Vector so in this case our black Vector is our resultant vector vector subtraction is almost as easy if we think of algebraically adding two values a plus b equal C well if we want to subtract them a minus B is equivalent to writing a + b so if this is Vector a and this is Vector B how do we get neb we just switch the direction of the vector now we have a and negative B the light blue Vector is now pointing in the opposite direction so we have a + b which is the same as a minus B now it's an addition problem where we're adding A and B how do we add vectors we line them up tip to tail so let's slide this a vector over here so that it's lined up tip to tail with b and then we can draw a line from the starting point of our first Vector to the ending point of our last Vector to give us our answer this black Vector straightforward once again basic Vector manipulation now when we're talking about vectors often times you have vectors at an angle and if we draw this on an XY AIS we can label the angle between the horizontal axis and our Vector as angle Theta well there are many times in physics where we can like make our life much much much simpler if we only work in one dimension at a time so we could think of this gray Vector as actually being comprised of the addition of two smaller vectors one vector which travels here along the horizontal axis we will call the X component of the vector the other component is the vertical component and if we add those two you can see that our gray Vector is the resultant of our two yellow vectors our two Vector components which lie only along a single axis this is our X component of vector a this is our y component of vector a and we can use basic trigonometry to figure out exactly how big those component vectors are ax because it's the adjacent side is going to be equal to the magnitude of vector a times the cosine of angle Theta the Y component of our Vector a y is going to be equal to a and because now we're talking about the side of the right triangle that is opposite angle Theta that's going to be a sin Theta and you could actually go in the other Direction too if you happen to know a y and ax you or a any two of those three sides you could always go and find that angle using trig 2 for example if we knew the vector components ax and a y and wanted to find Theta we know that the tangent of theta is the opposite side a y over the adjacent side ax therefore angle Theta would be the inverse tangent of a y over ax let's take a look at an example if a soccer player kicks a ball with an initial velocity of 10 m/s at an angle of 30° above the horizontal find the horizontal and vertical components of the ball's velocity well let's draw our axes here first there's our y AIS and our x-axis and the soccer player kicks the ball with an initial velocity so there's our initial velocity Vector we'll call that VI for initial velocity which has a magnitude of 10 m/s and it's at an angle of 30° above the horizontal we want to find its horizontal and vertical components well to find its horizontal component here ax that's going to be a CO Theta or 10 m/s now this that I'm substituting in with units 10 m/s time the cosine of 30° plug that into my calculator and I get something right around 8.7 m/s so we found its horizontal component now let's see if we can't find its vertical component a y its vertical component will a sin Theta once again I substitute in withd units 10 m/s sin 30° or 5 m/s and if we look at this just to see if this makes sense we can see that our X component is bigger than our y that makes sense the green Vector on our drawing is bigger than the orange vector and we could even check this out using the Pythagorean theorem a a^2 + b^2 equal our hypotenuse squar right so if we took 8.7 s added it to 5^ 2 we should come up with 10^ SAR which would be 100 looking at another example here we're talking about an airplane that flies with a velocity of 750 km hour at an angle 30° south of East What's the magnitude of the planes Eastward velocity well let's start off with the diagram if we draw our axis our Compass there's North South West and East its velocity is 750 km per hour but it's 30° south of East so if we start East and go 30° South that angle must be 30° and this must be 750 km per hour we we want to find the magnitude of the plane's Eastward velocity well if we want its Eastward velocity we want this component here along the x-axis that's its Eastward component so we're looking for ax which must be a cine Theta because that's the adjacent side of our triangle or 750 kilm per hour times the cosine of 30° plug that into my calculator should get something close to 650 kilm per hour let's take a look at yet another one here we have a dog walking a lady a very excited dog at about 8 me a distance of 8 M due north I guess that would be a displacement a change in position with Direction so the dog walks our lady 8 m due north and then 6 M due east 8 m North 6 M du East determine the magnitude of the dog's total displacement well to do that this looks like a vector addition problem we start at our starting point our vectors are already lined up tip to tail so we go from the starting point of our first to the ending point of our last that red Vector must be our dog's total displacement how do we figure out how big that is well we could use the Pythagorean theorem because that's a right triangle a 2 + b^ 2 = our hypotenuse squar or 8 m squar + 6 m squar equal our hypotenuse squ or 100 square m equals our hypotenuse squar take the square root of both sides and we find out our hypotenuse or total displacement must be 10 m now we've talked about the resultant Vector being the sum of two Vector two or more vectors let's talk about the equilibrant vector the equilibrant vector is the opposite of the resultant to find an equil equilibrant Vector all we do is we find the resultant first and then we switch its direction so if you had a problem like this where it says the diagram below represents two concurrent forces concurrent just means they're happening at the same place at the same time draw the equilibrant force Vector well to find the equilibrant let's first find the resultant these aren't lined up tip to tail so I'm going to redraw my vectors so that they are lined up tip hip to tail and to find the resultant I start by drawing a vector from the starting point of the first to the ending point of the last now because I don't want the resultant I want the equilibrant all I do is I switch the direction of that Vector right now my resultant is pointing in that sort of Northeast Direction so if I want the equilibrant I need to switch its direction my equilibrant Vector must look something like this there's my answer so next steps in your house or backyard Define a starting point and an ending point somewhere in your yard or in your house then find three vectors that if you followed all three vectors would get you from your starting point to the ending point and make sure you write those down with their size how far and their Direction then try rearranging those three vectors and see if you get to the same point each time does that work and can you explain why or why not if you need any extra help check out A+ physics.com thanks and have a great day