Question

Galileo's great-great-great grandchild stands at the top of a vertical tower 57.8 m tall with a Chianti bottle. How long does it take for the bottle to fall to the ground, if it was just dropped from the tower? How far does the bottle land from a point on the ground directly beneath the point from which it was launched, if the bottle was thrown straight out horizontally from the tower with a speed of 12.9 m/s? What is the bottle's horizontal component of velocity, if the bottle was thrown straight out horizontally from the tower with a speed of 12.9 m/s? What is the magnitude of its velocity just before it strikes the ground, if the bottle was thrown straight out horizontally from the tower with a speed of 12.9 m/s?

Answer 1

Given:

Height of the tower H = 57.8 m

(a)

from the kinematic relation

s = ut+1/2at² ....... (1)           (s = H = 57.8 m )

a is the acceleration due to gravity (9.8)

t = sqrt (2s/a)

= sqrt(2*57.8m/9.8m/sec²)

= 3.43 sec

(b)

final velocity of the when touches the ground

v = sqrt (2gh)

   = sqrt[(2)*(9.8m/sec²)*(57.8m)]

   = 33.65 m/sec

(c)

now in this case the bottle is treated as a projectile .

so initial velocity of the projectile u = 12.9 m/sec

Range r = u*t

             = (12.9 m/sec)(3.43 sec)

             = 44.249m

(d)

Horizontal component of velocity is constant at any point of path of the projectile so

horizontal component of velocity = 12.9 m/sec.

(e)

horizontal velocity u = 12.9 m/sec

vertical velocity v = 33.65 m/sec

resultant velocity at the point where it touches the ground

V = sqrt [(12.9)²+(33.65)²]

    = 36.037m/sec.