Question

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?

Answer 1

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

Answer 2

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

Answer 3

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