resisted motion (1 Viewer)

[doiyoubi]

a particle of mass m is projected vertically upwards with velocity u in a medium whose resistance R is mkv where v is the velocity and k is a constant. After a time t the velocity is v at a height y above the point of projection. prove that:
y=[(u-v)/k]-[g/k²]ln[(g+ku)/(g+kv)]

 

[slee89]

v dv/dy = -kv-g
dv/dy = -kv-g/v
dy/dv = v/-kv-g
y = - ∫(v/kv+g) dv
= -1/k∫(kv+g-g/kv+g)dv
=-1/k∫(1-g/kv+g)dv
=-1/kv+g/k^2∫(k/kv+g)dv
y = -1/k+g/k^2ln[kv+g] + c
y = 0, v = u
0 = -u/k+g/k^2ln[ku+g] + c
c = u/k - g/k^2ln[ku+g]
y = -v/k-g/k^2in[kv+g]+u/k-g/k^2ln[ku+g]
y = -g/k^2ln[ku+g/kv+g] + u-v/k