. How tall is the building?

Question

A ball is dropped from the roof of a tall building. When you start your stopwatch, the ball has
already reached half the height of the building. You stop your stopwatch exactly before the ball hits
the floor. The stopwatch shows one second. How tall is the building?

Answer 1

Since the ball was released from rest, it reached h/2 in time t that satisfies

 

h/2 = gt²/2 ⇒ h = gt²

 

In 1 more second it has reached a distance h from the top.

 

h = g(t + 1)²/2 = g(t² + 2t + 1)/2

 

You have 2 equations in 2 unknowns h and t:

 

h = gt²

h = g(t² + 2t + 1)/2

 

Subtract top equation from bottom equation and get

 

-gt²/2 + gt + g/2 = 0

 

t² - 2t - 1 = 0

 

Solve this quadratic equation for t, and plug t into either of the 2 equations to find h.

 

t ≅ 2.414 sec

 

h = gt² = 9.8(2.414)² m ≅ 57.11 m

 

It should give the same answer plugging t into

 

h = g(t + 1)²/2 = 9.8(1 + 2.414)²/2 m ≅ 57.11 m