Question 823176
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a partial answer:
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equation of ballistic motion:
h(t) = -1/2gt^2 + v0t + h0
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on planet earth at the surface of the planet:
given:
g = 10 m/ss
h0 = 1.25 m
t = 2.5 sec
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h(2.5) = -5(2.5)^2 + v0(2.5) + 1.25 = 0
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-5(2.5)^2 + v0(2.5) + 1.25 = 0
v0 = ( 5(2.5)^2 - 1.25 )/2.5
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answer1:
speed Nathan hits the ball = v0 = 12 m/s
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for this problem:
h(t) = -5t^2 + 12t + 1.25 = 0
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the above quadratic equation is in standard form, with a=-5, b=12, and c=1.25
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to solve the quadratic equation, by using the quadratic formula, copy and paste this:
-5 12 1.25
into this solver: https://sooeet.com/math/quadratic-equation-solver.php
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this quadratic has two real roots (the x-intercepts), which are:
t = -0.1
t = 2.5
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the quadratic vertex is a maximum at: ( t= 1.2, h(t)= 8.45 )
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negative time doesn't make sense for this problem, so use the positive root
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answer2:
the ball reaches a max height of 8.45 m (1.2 secs after being hit)
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answer3:
the ball touches the ground 2.5 seconds after being hit (this is given but confirmed by the positive root)
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Solve and graph linear equations:
https://sooeet.com/math/linear-equation-solver.php
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Solve quadratic equations, quadratic formula:
https://sooeet.com/math/quadratic-formula-solver.php
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Solve systems of linear equations up to 6-equations 6-variables:
https://sooeet.com/math/system-of-linear-equations-solver.php