WEBVTT
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we know that the range equation this is B for
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part of the range equation is equaling V initial squared
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sine of to theatre divided by G. And so
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we have essentially two parts to one one trajectory.
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So the one trajectory is letting it bounce, setting
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the baseball bounce. And so we can say that
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this would be our prime. This would be equaling
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the initial squared sine of tooth ada over G for
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the first for the first throw. And then after
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it bounces, it bounces with half of the same
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velocity. So we can say that this would be
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plus the initial over to quantity squared sine of tooth
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ada over G. So that would be the full
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range given that it bounces. And then we can
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say that if it doesn't bounce, are is simply
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equaling the initial squared over G because we're trying to
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compare this to an angle where where this angle is
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45 degrees. So now, with this angle equaling
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45 degrees, we can set these two equal to
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one another because supposedly this is having the same range
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. So we can say that then the initial squared
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over G is going to be equaling the initial squared
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sign of tooth Ada, divided by G plus the
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initial squared sine of tooth ada divided by four G
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. And this is this would essentially describe the trajectory
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of the ball as it bounces one time with one
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bounce. And then this would be if it was
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, if the range which maximized where fada here.
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Fate as equaling 45 degrees. So this is the
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maximum range, and then we want to find the
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angle at which you should throw the ball in order
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to get to the maximum range. However, we
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want to get there faster. So we would set
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these equal to one another. And we can then
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say that V initial squared over. Chief is equaling
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the initial squared over G. This would be sign
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of two fada plus sign of tooth Ada over four
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. And so we can then say that four part
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eh, one is going to be equaling Sign of
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tooth ada, uh, multiplied by one plus 1/4
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and this is equaling five over four. Sign of
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tooth data. And so Fada is gonna be equaling
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. This would be 1/2 times art sign of four
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over five. And so this is Seita is equaling
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26.6 degrees. So this would be your answer for
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party. That ball should be thrown at an angle
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of 26.6 degrees above the horizontal so that it can
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reached the maximum range, so this would be above
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horizontal, so ball reaches max range. Uh huh
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. Fada equals 45 degrees. So this would be
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your answer for party. And then for part B
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, we can say the time taken for no bounce
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would be t And so the time would be to
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be a sign of Fada over G. Of course
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here Seita is equaling 45 degrees. In this case
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, so T is going to be equaling two times
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the initial velocity over G times sign of 45 degrees
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. And so this is gonna be equaling. We
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could say 1.41 times the initial velocity over, Chief
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. Now, this would be for one bounce for
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sorry. This would be for no bounce for one
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bounce. We can say that. Then this would
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be t prime and t prime is gonna be equaling
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the time for the first throw freely through the initial
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throw and then the time after the first bounce.
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And so we can say that t someone is gonna
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be equaling to the sign of data over G.
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And so this would be equaling two to v sign
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of 26.6 degrees over G. And we can say
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that then t's of one is equaling. This would
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be to the over G multiplied by 0.448 Now t's
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up to is after the second. If that's where
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the first bounce. So that one bounce. And
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so this would be two times the initial velocity divided
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by two. Because only after the first bounce it's
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only moving with half of its initial velocity times sign
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again, 26.6 degrees over G. And so we
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can then say that this would be equaling two 0.448
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once again, but without the factor of two.
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So it's multiplied by the initial developed by G.
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And so now we can say that T crime is
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gonna be equaling two chiefs of one plus t's up
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to, which is simply gonna be equal in 1.34
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times the over G or essentially two times 20.448 plus
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0.448 So three times 30.448 which is 1.34 multiplied by
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the initial divided by G And so dividing these to
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weaken say T prime over T, this would be
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equal in 1.34 the initial over G divided by 1.41
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the initial over G, and we find that four
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part B T prime over tea is equaling 40.95 So
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this means that the time it takes for the ball
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to reach a certain horizontal range is 95% of the
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time taken if the ball was thrown without any bounce
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. So this is one bounce, and this is
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again no bounce. So, as you can see
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, throwing it at a short of the smaller angle
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, 26.6 degrees is actually having it, having the
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baseball get to its destination faster. Then, if
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you threw it without any bounce, so even though
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after the bounce it loses initial, it loses its
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velocity by half, it still gets there faster.
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It's only 95%. It'll be 95% of the time
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taken, um, without any bounce so this would
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be our final answer. So it is true in
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the sense that baseball players allow the ball base.
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Probably baseball players allow the ball to bounce once so
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that it will get to its destination a bit quicker
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. That is the end of the solution. Thank
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you for watching.