Free fall of stone - physics problem and solution

A stone is dropped from a bridge 100m above the water. How long does it take to fall (1) the first 50, (2) the second 50m?

Given:

H = 100m

D₁ = 50m

D₂ = 50m

We are looking for:

t₁ = ? – time of falling the first 50m,

t₂ = ? – time of falling the second 50m

Solution:

Using the formula for motion along a straight line with constant acceleration we write (1) this is the formula for the first 50 m

The second 50 m was traveled with initial velocity v₀, so equation will be

(2)

The initial velocity v₀ for the second 50m is the final velocity acquired after traveling first 50m. From (1) we find

(3)

so the velocity after moving (falling in our case) during t₁ with gravitational acceleration g will be

(4)

Substituting equation 4 into equation 2, after a minor rearrangement we get

(5)

and this is a standard quadratic equation of the form

ax² + bx + c = 0 (6)

where

, , and

Standard procedure of solving equation 5 gives two roots

(6) and

(7)

Formula (6) will give a negative value of time (it is possible in algebra, but not in physics) so this solution must be rejected.

Substituting numbers given in the problem into Equations (3) and (7) we get

t₁ = 3.193s, t₂ = 1.323s.