Question

A flea can jump a vertical height h. What is the maximum horizontal distance it can jump? What is the time in the air in both cases?

Answer 1

maximum range is at 45 degrees launch from horizontal if there is not air friction.
(1/2) m V^2 = m g h
so V = sqrt (2 g h)
that is your launch speed
u = V cos 45, horizontal speed,constant
Vi = V sin 45 = initial vertical speed
range = u t where ti is time in air
need to get time in air
v = Vi-g (time up)
0 = Vi - g (time up)
time up = Vi/g
t in air = 2 * time up
= 2 Vi/g
so
range = u *t = V cos 45 * 2 V sin 45 /g
= 2 sin 45 cos 45 V^2/g
but V^2 = 2 gh
so
range = 2 sin 45 cos 45 *2h
sin 45 = cos 45 = 1/sqrt 2
so range = 2 h