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A ball is shot into the air from the edge of a building

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50 feet above the ground

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It s initial velocity is 20 feet per second

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The equation h (height) = -(negative)16t squared+20t+50 can be used to model the height of the ball

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after t second

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And I think in this problem they just want us to accept this formula

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Although we do derive formulas like this

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we show why it works for this type of problem in the Khan academy Physics playlist

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But for here we will just go with the flow in this example

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So they give us the equation that we use to model the height of the ball after t seconds

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and then say about how long does it take the ball to hit the ground

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So if this is the height, the groudn is when the height is equal to zero

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So hitting the ground literally means that h (height) is equal to zero

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So we need to figure out at which times does h (height) equal zero

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So we are really solving the equation

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0 = -16tsquared + 20t + 50

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And if you want to simplify this a little bit

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Let's see everything here is divisible at least by 2

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so let's divide everything by negative 2 just so that we can be rid of this negative coefficient

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So you divide the left hand side by negative 2, you still get a zero

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Negative 16 divided by negative 2 is eight so 8t squared

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20 divided by negative two is negative 10

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Minus 10t

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50 divided by negative 2 is minus 25

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is equal to zero. If you okay with this on the left hand side you can put it on the left hand side

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We can say this is equal to 0

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And now we solve

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And we can complete the square here or apply the quadratic formula which is derived from

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completing the square

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and we have this in standard form, we know that this is our a

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this our b and this over here is our c

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And the quadratic formula tells us that the roots

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In this case in terms of the variable t are going to be equal to

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negative b plus or minus the square root of b squared minus four a c.

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All of that over 2 a

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So if we apply it we get t is equal to negative b

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b is negative 10

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so negative negative 10 is going to be postitive 10

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plus or minus the square root of negative ten squared

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well thats just positive 100 - 4*a which is 8

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times c which is negative 25

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and all of that over 2 a

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a is 8 so 2a is 16

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and this over here we have a negative sign

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negative times a negative... so this is a positive

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4 times 25 is 100

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100 times 8 is 800

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so all that simplifies to 800

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and we have 100 + 800 under the radical sign

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so 10 plus or minus the square root of 900 all of that over 16

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is equal to 10 + or - 30 over 16

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so we get time is equal to 10+30/16

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40-16 which is the same thing if we divide the numerator

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and denominator by 4

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which is 10 over... or even better divide by 8

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5 over 2

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so thats one solution.

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Thats if we add the 30

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if we subtract the 30

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so or t = 10 -30 which is -20 over 16

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divide the numerator and the denominator by 4

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and you get -5 over 4

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Now we have to remember

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were trying to find a time

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and so a time at least in this problem

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is positve

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we want to figure out how long it takes for the ball

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to hit the ground

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we dont want to go back in time, we dont want negative time

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we only want to think about our positive answer

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and so this tells us that the only root that should work

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is, we have to assume that this is in seconds

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so 5/2 seconds

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i woudnt worry to much about the phisics here

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i think they just want us to apply the quadratic formula

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into this modeling situation

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the physics, we go into a lot more depth in the physics playlist

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so lets verify that we definitley

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are at a height of 0 at 5/2 seconds

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so if t=5/2 then this expression is equal to 0

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so we have, lets try it out, so we have

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-16*5/2^2

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+20*5/2

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+50

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this needs to be equal to 0

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so this is -16*25/4 + 50+50=0

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-16/4 is -4

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4/4 is 1

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so -4/25 which is 100, +50

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sorry, -4/25 is NEGATIVE 100

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100+50+50

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so we have -100 plus 100

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so thats definatly going to be 0

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we get 0 = 0

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and it all checks out

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we hit the ground after 5/2 seconds

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or another way to think about it is 2.5 seconds