in this problem we've got a golf ball being launched from ground level i'm going to put the
origin right where the ball starts so that's the point at which x and y are both equal to 0. i'm
given an initial speed on this launch of 45 meters per second and an angle of inclination of 52
degrees so before i start doing anything i'm going to break the velocity into components so that
vertical component just comes from translating this vertical leg of the right triangle that
you can break the original velocity into so in other words that magnitude is equal to the
magnitude of the of the dashed line that i just drew so my x velocity component that's v naught x
well that corresponds to the adjacent side of this thing so it's going to be hypotenuse
times the cosine of the angle 45 cosine 52 degrees and when i run the numbers on this i've got 27.7
meters per second the y component corresponds to the vertical leg of that right triangle and so i
get v naught y is equal to 45 sine of 52 degrees using the sine function because i'm trying to
grab the opposite side of the right triangle to the angle that was labeled and i get 35.5
meters per second for the y component of the velocity in part a i'm asked to get the flight
time for the projectile and so the way to get a handle on this is to ask what do i know at the end
of the flight is it something about x or something about y and in this case it's something about y
y is equal to 12 meters i'm told that this is an elevated surface 12 meters above the
starting point so given that i have this little bit of extra information about why that's how
i'm going to attack the flight time problem now remember the equations of motion for the vertical
and horizontal directions can be unlinked and just written down separately in the y direction there's
an acceleration of negative g in the x direction there's no acceleration at all so i'm going to
take my standard kinematics formula but instead of plus one half a t squared at the end i'm going
to plug in negative g for the acceleration and there's my starting point now my final y position
is 12. my initial was zero my initial y velocity probably should have labeled it that way is 35.5
the flight time is unknown plug in g equals 9.8 and we have a quadratic equation to solve so i'm
going to move all the terms to the left-hand side mostly because i'd like to have a positive 4.9
i like to have a positive a for the quadratic formula my preference is to just use technology
to grind through this and find the value of t however sometimes we're required to do this by
hand so i'll illustrate what that looks like so i'm appealing to the quadratic formula here
t is negative b that's 35.5 plus or minus square root b squared 35.5 squared minus 4 times a
times c all divided by twice a which is 9.8
and i'll simplify the square root down to a decimal and i get 32.0 for that so 35.5
plus or minus 32.0 all divided by 9.8 so we'll go ahead and plug in the minus
sign version and the plus sign version and simplify them both down to a number
i'm doing the minus first 35.5 minus 32 all divided by 9.8 and i've got a time of 0.357
seconds then i plug in the plus version 35.5 plus 32 divided by 9.8 and i've got 6.89 seconds
so i have to ask which one of these is the one i'm going to keep both of them are physical
solutions they both happened in reality and i have to think like what did i plug into the
mathematics here and what i plugged in was a 12 for the height and i said tell me what time it is
and by the time i solve the equations i get two times the reason why is because back here
the golf ball was at a height of 12 meters on its way up so that's all this small time
solution means but we're not interested in it the flight terminates at 6.89 seconds finally we're
asked to get the range for the projectile that means the horizontal displacement for the flight
and this part is simple by comparison the x motion of a projectile is just a constant velocity motion
because there's no forces acting left or right on it only straight down from gravity
so there's no x acceleration at least when we're ignoring the effects of air resistance
my initial x position was zero my initial x velocity i've already taken care of that was 27.7
meters per second i've got the flight time 6.89 seconds and i come out with a range of 191 meters
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