Question 1047694: If a body travels half its total path in the last 1.70 s of its fall from rest, find the total time of its fall (in seconds).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
the equation for distance of a free falling object is:
d = g * t^2 / 2
d is the distance traveled.
g is the acceleration due to gravity which can be rounded to 9.8 meters per second squared.
t is the time in seconds.
let d1 = g * t^2 / 2
let d2 = g * (t + 1.7)^2 / 2
since d2 is the total distance traveled, then d2 must be equal to 2 * d1.
your equation becomes:
d1 = g * t^2 / 2
2 * d1 = g * (t + 1.7)^2 / 2
since d1 = g * t^2 / 2, then 2 * d1 is equal to 2 * g * t^2 / 2
you have 2 * d1 = g * (t + 1.7)^2 / 2 and you have 2 * d1 = 2 * g * t^2 / 2
this means that:
2 * g * t^2 / 2 is equal to g * (t + 1.7)^2 / 2
simplify this to get:
g * t^2 = g * (t + 1.7)^2 / 2
simplify further to get:
g * t^2 = g * (t^2 + 3.4 * t + 2.89) / 2
multiply both sides of this equation by 2 to get:
2 * g * t^2 = g * (t^2 + 3.4 * t + 2.89)
divide both sides of this equation by g to get:
2 * t^2 = t^2 + 3.4 * t + 2.89
subtract all terms on the right side of the equation from both sides of the equation and you will get:
2 * t^2 - t^2 - 3.4 * t - 2.89 = 0
combine like terms to get:
t^2 - 3.4 * t - 2.89 = 0
factor this quadratic equation to get:
t = -.70416 or t = 4.1042
t can't be negative, so t = 4.1042 seconds.
that is the time it took for the first half of the distance.
the time it took for the second half of the distance is 1.7 seconds.
the time it took for the total distance is therefore 5.8042 seconds (4.1042 + 1.7).
to see if this is correct, substitute in the original equation.
d1 = g * t^2 / 2 becomes d1 = 9.8 * 4.1042^2 / 2 = 82.53784244 meters.
d2 = g * (t + 1.7)^2 / 2 becomes d2 = 9.l8 * 5.8042^2 / 2 = 165.0748144 meters.
it traveled 82.53784244 meters in 4.1042 seconds and it traveled 165.0748144 meters in 5.8042 seconds.
in the last 1.7 seconds of its fall, it traveled 165.0748144 - 82.53784244 meters which is equal to 82.536972 meters.
that's very close to 82.53784244 meters.
the difference is due to rounding.
a more exact definition of the acceleration due to gravity is 9.80665 meters per second squared.
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