Free Fall Exercises
Exercise Two, Calculations Combining Free Fall and Sound Wave Velocity.- A stone is dropped from the top of a building. The sound of the stone hitting the ground is heard 6.5 s later. If the sound velocity is 340 m/s, calculate the height of the building.
Solution.- Calling t1 the time it takes the stone to reach the ground on free fall and t2 the time it takes the sound wave to travel from the ground to the top of the building, then t1+t2 = 6.5 s
The height h of the building is obtained from
the general formula
y=yi + vit + ½ at2, with
y=h, vi=0, a=9,8 m/s2.
Then h=½ gt12=½ ·(9.8 m/s2)t12=(4.9 m/s2)t12
Also, h = (340 m/s)·t2. Equating
the two equations and replacing t2 by (6.5 - t1) to
(4.9 m/s2)t12= (340 m/s)·t2 = (340 m/s)·(6.5 - t1) or 4.9·t12= 2210 - 340t1, with t1 in seconds,
4.9·t12 + 340t1 - 2210 = 0
The solution of this quadratic equation is 5.98 s.
Then, the height of the building is h = (4.9 m/s2)(5.98s)2 = 175 m
Exercise Three. Free Fall Combining Initial Velocity and Time to Reach Ground.- A man standing at the top of a building throws a ball vertically upward with a velocity of 14 m/s. The ball reaches the ground 4.5 s later. a) What is the maximum height reached by the ball? b) How high is the building? c) With what velocity will it reach the ground?
a) Maximum height means v = 0. Then using v = vi + at, and a vertical axis with origin at the top of the building pointing upward, with
v = 0, a = -9.8 m/s2 and vi = 14 m/s, the time to reach maximum height is t = (14 m/s)/(9.8 s) = 1.43 s
Now, using y = yi + vit + ½ at2 with the same vertical axis already mentioned, yi = 0, vi = +14 m/s, a = -9.8 m/s2, we get
ymax = (14 m/s)·(1.43s) - (4.9m/s2)(1.43 s)2 = 10 m
b) Keeping the same axis as before, and using vi = +14 m/s, a = -9.8 m/s2 and t = 4.5 s in y = yi + vit + ½ at2, y = (14 m/s)·4.5 s - 4.9m/s2(4.5 s)2 = -36.2 m. The result is negative as expected since the ground is in the negative side of the axis. The height of the building is then 36.2 m.
c) The velocity reaching the ground is v = 14 m/s - 9.8 m/s2·4.5 s = -30.1 m/s.
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