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?
Answers
Damon
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
(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