Question

urgent ------> Q. definition of displacement and distinguish between uniformly and non uniformly accelerated motion

Answer 1

Displacement is the shortest distance between the initial and final position of a body. It is a vector quantity, meaning it has a direction associated with it.

Uniformly accelerated motion: The acceleration is constant which means that equal increments in velocity in equal interval of time.
$\frac{dv}{dt} = constant = a$
Hence v = at + c, where c is some constant depending upon the initial velocity of the body and t is the time
The displacement is given by:
$\frac{d^2x}{dt^2} = constant = a\ (assumed) \\ \frac{dx}{dt} = at+c_1 \\ x = at^2 + c_1t + c_2$, where c1 and c2 are constants
Summary = Velocity profile is linear and displacement profile is parabolic.

Non-uniformly accelerated motion: The acceleration is variable which means that acceleration itself is a function of time
$\frac{dv}{dt} = a(t)$, a(t) denotes acceleration as a function of time
Hence v = $\int\limits^{t_2}_{t_1} {a(t)} \, dt$
The displacement is given by:
$\frac{d^2x}{dt^2} = a(t)$
Summary = Velocity profile is at least parabolic and displacement profile is at least cubic.

Velocity and displacement profiles are based on the assumption that acceleration is a polynomial function of time.