Galileo\'s great-great-great grandchild stands at the top of a vertical tower 57

Question

Galileo's great-great-great grandchild stands at the top of a vertical tower 57.4 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? What is the velocity of the bottle as it hits 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 13.1 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 13.1 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 13.1 m/s?

Explanation / Answer

Given that height of the tower ,h is 57.4 m.

As intial velocity is 0, we can use second law of motion to find out time taken by bottle to hit the ground.

h = ut + 1/2 at^2

57.4 = 0 + 1/2 * 9.8 * t^2

t = 3.42 sec

so velocity when it hits the ground is

v = u + at

v = 0 + 9.8 * 3.42

v = 33.5 m/s

If we give an intial horizontal velocity 13.1 m/s then distance from point on the ground directly beneath as there is no force in horizontal direction is
d = 13.1 * 3.42

d = 44.8 m

As there is no force in horizontal direction therefore no acceleration, hence it will travel with constant horizontal velocity component. Therefore

horizontal velocity component = 13.1 m/s

Magnitude of velocity just before it hits the ground is

v = sqrt((33.5)^2 + (13.1)^2)

v = 35.97 m/s

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