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
To find the speed of the car, we need to use the concept of power. Power is defined as the rate at which work is done or energy is transferred. In this case, since the car is being driven with constant power, we can set up the equation:
Power = Work / Time
Since the car starts from rest, we can assume that it accelerates uniformly. In this case, the work done is equal to the change in kinetic energy of the car.
Work = Change in kinetic energy = (1/2) * m * (v^2 - 0^2) = (1/2) * m * v^2
Now, let's substitute this into the power equation:
Power = (1/2) * m * v^2 / Time
To find the speed of the car, we need to rearrange the equation to solve for v:
v^2 = (2 * Power * Time) / m
Finally, taking the square root of both sides:
v = sqrt((2 * Power * Time) / m)
Now, to find the speed of the car, you need to know the mass of the car (m), the power applied to drive the car, and the time it takes to cover the distance (x). Plug these values into the equation, and you will be able to calculate the speed of the car.