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2015-06-20T14:36:11+05:30

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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.



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