physics
posted by muhammad .
a car of mass 'm'starts from rest and is driven with constant power.after some time it covers a distance 'x'.what should be the speed of car?how we can find the speed of car?

power = force * speed
p = F v
so
F = p/v
F = m a = p v
so
a = (p/m) v
or
dv/dt = (p/m) v
try form
v = b e^kt + c
dv/dt = b k e^kt
so k = p/m
so
v = b e^(p/m)t + c
at t = 0, v = 0
0 = b + c
so
v = b e^(p/m) t  b
at time t, it is at x
this requires solving differential equation of form
dx/dt = b e^kt  b
x = (b/k) e^kt  bt + c
at t = 0, x = 0
0 = (b/k) + c
so c = b/k = bm/p
x = (bm/k) e^(p/m)t  b t bm/p
given an x at time t, solve for b 
power = force * speed
p = F v
so
F = p/v
F = m a = p/ v
so
a = p/(m v)
or
v dv/dt = (p/m)
v dv = (p/m) dt
v^2/2 = (p/m) t + constant which is zero
v^2 = (2p/m)t
v = (2 p t/m)^.5 
dx/dt = (2 p /m)^.5 t^.5
let k = 2 p/m
dx/dt = k t^.5
dx = k t^.5 dt
x = (k/1.5) t^1.5 + c
when t = 0, x = 0 so c = 0
x = (4 p/3m) t^1.5
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