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Putting it all together: more fun with vector components we say fun with

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vector component that's like fun....

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all right story time yeah I'll do that

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okay time to take a break from vectors and hit the slopes just glide down a [Man skiing on the slopes]

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mountain letting gravity do all the work moving horizontally and vertically in a

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diagonal direction, uh-oh sounds like a two-dimensional vector doesn't it make

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sense skiing is a lot like that experiment Galileo did the one where he [Galileo skiing down a mountain]

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discovered the force of gravity by rolling a ball down a ramp a bunch of

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times except with skiing we don't have some old Italian guy timing us with a

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water clock and like we said gravity will be the force that's sending us

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downhill sadly our jet-powered skis haven't been [Man with jet-powered skis appears]

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delivering us yet so there won't be any horizontal acceleration just vertical

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however because we'll be going down on an incline aren't we're all velocity

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magnitude will be increasing but hold on we're really awesome at skiing so we'd [Man performs ski jump]

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be doing all sorts of radical jumps and stuff right and it would throw off all

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the math so instead we'll be observing the motion of our buddy Rex

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Tyrannosaurus Rex yeah he's a newbie to this winter wonderland so he'll just [T-Rex skiing down mountain]

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hope to stay upright.... so let's say t-rex is going down a mountainside and

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for the sake of simplicity we'll say this is a completely frictionless

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ice-covered hill and we'll put the hill in an angle of 40 degrees you know what

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Rex's speed went from 200 meters from the ski lift, first things first we

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know the total distance along this diagonal yeah Rex just keep skiing

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and we know the angle of incline well do we have enough to calculate the velocity

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well the best equation to use would be this one it tells us that the square of [Equation of velocity appears]

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the final velocity equals the square of the initial velocity plus two times the

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acceleration times the distance well the initial velocity will be zero and the

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distance will be 200 meters but we don't have the acceleration let's take care of

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that problem right now as the Italian stallion

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Galileo himself taught us acceleration down an incline equals the force of [Galileo appears galloping on a horse]

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gravity times the sine of the angle of inclination well the acceleration of

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gravity is 9.8 meters per second squared just like always when we multiply that

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times the sine of 40 degrees we get an acceleration of six point three meters

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per second squared which is a lot of acceleration for a nine-ton reptile okay

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now we can use that velocity equation we looked at before since the initial

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velocity is zero the square of the final velocity equals two times the

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acceleration times the distance well doing that math gives us two thousand

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five hundred twenty meters per second ah but we need to find the square root of

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that number to be able to find the final velocity while taking the square root

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gives us a final velocity of fifty point two meters per second but there might be

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a little problem remember when we said Rex was new to this whole skiing thing [People skiing and T-Rex appears]

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and that he's super big yeah he's not gonna be stopping anytime soon in fact

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it looks like he's gonna go over the edge of the mountain which would be [T-Rex falls off edge of mountain]

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about 12 meter fall before he hits the ground below so what's the speed at the

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bottom of this tumble well in this case the initial velocity is definitely not

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zero after all we just figured out that our big friend was going fifty point two

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meters per second and since this is a vector quantity it needs a direction as

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well that direction was 40 degrees but we need to break that velocity down into

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its vertical and horizontal components and yep we're getting triggy with it all

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right so if we put this on a graph we can see that we have a right triangle

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here and the velocity that we calculated is the hypotenuse of that triangle [Triangle appears on graph]

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hypotenuse that thing right there since we know the incline angle we can use

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trig functions to find out the magnitude of the component velocities which make

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up the other sides of this triangle, time for for sohcahtoa

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...... it's not an island in the South

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Pacific it's how we remember our primary trig functions well the soh when

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sohcahtoa reminds us that the sine of an angle equals its opposite side divided

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by the hypotenuse so the sine of 40 degrees equals v sub y over 50 point two

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meters per second when we multiply each side by fifty point two meters per

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second we find that the vertical velocity v sub y equals 32.3

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meters per second all right now for the horizontal velocity that makes up the [Triangle appears with horizontal velocity]

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adjacent side of our triangle which means we need to use the cah in

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sohcahtoa the cosine of an angle equals its adjacent side over the hypotenuse so

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once again we multiply both sides of the equation by the hypotenuse to solve for

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v sub x well that math we're doing here tells us that the horizontal velocity is

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38.5 meters per second okay we're making progress on this velocity stuff and

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Rex is making progress getting himself upright yeah [Rex attempts to stand up from ditch]

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those little arms aren't helping though all right now we can find the vertical

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velocity that Rex achieved right before he hit the ground to do that we'll use

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this equation again well this time we do have an initial velocity it's a thirty

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two point three meters per second we just figured out a minute ago and the

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acceleration will just be our good old 9.8 meters per second squared buddy

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there aka gravity, oh in the displacement well that'd be the twelfth meter plummet

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we plug in those numbers don't forget to find that square root and we get a final

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velocity of thirty five point eight meters per second and that's it all done

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we just need some hot cocoa and a paleontologist make sure Rex didn't break [Man with skis appears at lodge]

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any bones but oh wait there's one more things there always is we have our final

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vertical velocity and the horizontal velocity stayed the same

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until the Dino met the ground yeah but we need to figure out the combined

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velocity to get our actual final vector and we can do that with the Pythagorean

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theorem Pythagorean theorem that was

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Pythagoras' best theorem by far for this triangle v sub x squared plus v sub

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y squared will equal the square of the combined velocity so when we add the [Equations appear]

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squares and find that square root we get a final velocity 52.6 meters per second

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so Rex didn't pick up a ton of speed on his way down but it was only a 12 meter

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fall so not a big deal when we have a problem where the motion changes we need

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to be sure to break those motions into separate chunks then we just need to

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make sure we're using the right equation and that we're keeping track of any [Rex skiing down a slope]

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changes in the vector components like when Rex went off that cliff

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we had to factor in the vertical velocity he started with when he began

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as freefall to get the right combined velocity so make sure to take things

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step by step and pay attention to anything that needs to be carried over

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from the previous step all right let's go back to the lodge hope they have [man with skis appears at the lodge]

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those little marshmallows for the cocoa but Rex is still having a hard time

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trying to deal with all his you know equipment well looks like he's got his

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helmet balanced on top of one of his skis and he's trying to take those

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boots off at the same time if that ski tips so that one end is 60 centimeters

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lower than the other and the ski is 2 meters long well what are the final

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velocity x and y-components for the helmet when it slides down and off the

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ski ok oh this looks pretty familiar we just need to find the acceleration, well

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acceleration equals gravity times the sine of the theta angle but we don't

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know the degrees of that angle so more thinking, like we said before the sine of

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an angle equals its opposite over the hypotenuse when we have the length of

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the opposite side which is 60 centimeters and we have the hypotenuse

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which is 2 meters well to find the angle it's almost like we need to work

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backwards actually it's exactly like we need to work backwards like every trig [Rex skiing backwards]

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function has an inverse function so if the sine of x equals y the inverse sine

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of y equals x the inverse functions can look like this with a cute little

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negative 1 in superscript like that or you can add arc to the beginning of the

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function so the inverse of tangent is arctangent in the inverse of cosine is [inverse tangents appear]

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arc cosine and the inverse of sine is arc sine so let's make sure our

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calculator is set to use the inverse functions and that it's in degree mode [Rex using calculator]

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and let's put in those numbers all right the arc sine of 60 meters which we'll

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put in terms of meters over 2 meters equals 17 point five degrees now if you

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use that number to find the helmets acceleration as it slides down the ski [helmet slides down the ski]

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just isn't Rex's day so we'll go back to our acceleration equation acceleration

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equals gravity 9.8 meters per second squared times the sine of the theta

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angle which is 17.5 degrees okay so the helmet is picking up speed at a rate of

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2.94 meters per second too fast for a dinosaurs slow

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reflexes especially in the cold but remember we're trying to figure out the [Helmet hits man on the head]

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final component velocities here but to do that we need to find the combined

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velocity first we'll use this equation again and the initial velocity will be

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zero so we just need to calculate 2 times the acceleration times the 2 meter

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distance, O and find the square root so our combined final velocity is 3.4 [Combined final velocity equation appears]

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meters per second and that's it okay now we're in the homestretch we

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just need to break that vector down to its components it's important to

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recognize that earlier when we were figuring out the degrees of the incline we

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were working with lengths we weren't using velocity vectors just plain old

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scalar lengths so even though the triangle looks the same we're working on

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a whole new level well to find the component velocities we

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just need a little bit more trig and then we're done to find the horizontal

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vector we'll multiply our combined velocity times the cosine of the

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17.5 degree angle making the velocity in the x-direction

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3.27 meters per second and the vertical well, same basic idea except

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we'll be using the sine instead of the cosine which gives us a vertical

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velocity of 1.03 meters per second and yeah that makes sense since the [Skis propped up against a tree]

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angle isn't very steep most of the velocity is along the horizontal axis

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that's why the horizontal side of our triangle is a lot longer than the

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vertical these lines represent the magnitude of these vectors see all that [Boy and Rex sat at a table]

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works out okay now we're done for reals this time turns out all this ski stuff

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was just an excuse to do more math, in reality were terrible at skiing we do

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have a t-rex friend though....[Rex eats boy]

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