good day students in this clip we're going to learn how to solve projectile motion equations
which involves the
application of quadratic equations so let's go ahead and consider this question problem
one we have the following
situation and object is projected into the air with a path described by the quadratic
function h of t equals negative GT
Square was the 0T for speech of zero.
Where h of t is the height in feet at t seconds G is the force of gravity the sub
zero is the initial velocity and H subzero is the initial height babe ruth hit a baseball
that can be approximated by the function h(t)=-16t^2+64t+3.
Part a what is the maximum height of the baseball at what time does the ball which its maximum
height at what time does the baseball hit the ground justify your answer right let's
start with the first part part a it says what is the maximum height of the baseball H Max
can be found by evaluating age at Tmax Tmax is the time at which the ball reaches its
maximum height okay so how do we find Tmax Tmax is the same thing as the x-coordinate
of the vertex of the quadratic function okay so what is the formula for finding the x-coordinate
of the vertex use negative be over two a all right so the function under consideration
is h(t)=16t^2+64t+3 we can clearly see here that a is the coefficient of the
square -16 NBA is the coefficient of the key with first-degree term which is 64 remember
the order for quadratic equation in standard form this is a BC okay so these are the only
two we need to compute Tmax so Tmax into the x-coordinate of the vertex can be determined
by evaluating the expression -64 over two times -16 alive all I did is I substituted
a and b into this negative be over to a formula so using order of operations we evaluate the
denominator firstly have -64 over -32 -64 over -32 is to seconds so this is the time
at which the ball continues its maximum height but to want to look for the
maximum height also what was the value of the maximum height that H Max H so what is
Tmax here Tmax is to seconds so we're going to evaluate the function at T equals
two of our accident means whereby you have seen this equation here you substitute to
say to have -16 times two quantity squared +64×2+3 okay so use order of operations to
evaluate this expression right here alright we're going
to do next when first -16 disgrace for +64×2 is 128+3 and that is equal to -
64+131 okay and we combine this to you end up with 67 feet so this is the time we hit
the maximum height and this is the maximum height okay so let's combine the two answers
into one sentence that will answer the two-part question okay so the
baseball the baseball attains its maximum height its maximum height of 67 feet in two
seconds okay so that's that for the a part now let us shift our attention to the b part
it states at what time with the baseball hit the ground justify your answer okay so when
does the baseball hit the ground so the baseball let's write to that the baseball hits the
ground hits the ground when HD calls what West was the height when the baseball hit
the ground it is 0 feet from the ground okay so by setting off quadratic function equal
to zero and solving that will help to determine the time when the distance from the
ball from the ground is zero okay so what's the good of practically do is solve this equation
right here h of t we know what h of t is -16 T-square +6 4T +3 equals zero okay now let's
try and see if it can solve this by factoring that the
quickest way to so let's try the xgame and see if we can factor this ac goes on top AC
is -48 and b is 64 so are there any numbers that multiply to yield -48 and had to give
you 64 that is the question so let's go ahead and list all the possible teenagers that multiply
to give you -48 okay so we know that's 1×48 gives us 48 2×24 yields 48 3×16 yields 48
four times 12 yields 48 and 6×8 also yields 48.
So can we get a result of 48 by combining any of these two pairs the answer is absolutely
not this gives us 49 of 47 this gives us 26 or 22 this gives us a 19 or 13 this gives
the 16 and this gives us 614 or two so none of them gives us 48 so
what does that mean that means that this quadratic equation is prime so how do you solve a quadratic
equation that is prime because so that by either completing the square are using the
quadratic formula okay let's go ahead and use the quadratic formula after that is a
condensed version of completing the square X equals negative be plus or minus the square
root of the square -4 AC this entire expression over to a okay so let's list the value of
each variable right here eight is -16 B is 64 and see this three okay so we're going
to plug in all these values into the quadratic equation now the
variable our independent variable is time settee is going to take the position or take
the place of X all right so we have see equals -64+ or minus the square root of the square
which is 64 square -4 AC -4 times -16×3 this entire expression divided by 28 today which
is two times -16 okay so let's simplify the savings will have -64+ or minus dollars go
ahead and enter this expression in the calculator so I'm going to be using my TI 89 titanium
here so is going to have -64+ the square root of 64 square -4 times -16×3 okay I entered
the whole thing let me evaluate the radical peace first and get the
answer first let me make my calculator a little bit bigger L that is the
expression 4288 so for thousand 288 we know that two multiply by -16 is -32 so we have
to potential solutions here