Find the time of the fall?

Question

An object falls from a height h from rest. If it travels a fraction of the total height of 0.7578 in the last 1.00 s, find the time of its fall.

Find the height of its fall.

Answer 1

Let:

t be the time to fall from height h starting at rest,

t1 be the time to fall fraction (1 - f) of this distance starting at rest.

h = gt^2 / 2

t = sqrt(2h / g)

Similarly:

t1 = sqrt(2[1 - f]h / g)

The time in the last fraction f of the fall is:

t - t1 = sqrt(2h / g) - sqrt(2[1 - f]h / g).

Substituting t - t1 = 1.00 sec, f = 0.7578:

1.00 = sqrt(2h / g) - sqrt(0.4844h / g)

1.00 = sqrt(2h / g)[ 1 - sqrt(0.4844 / 2) ]

2h / g = 1 / [ 1 - sqrt(0.2422) ]^2

h = g / { 2[ 1 - sqrt(0.2422) ]^2 }

= 19.0 m.