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Physics Homework Help With Motion With Constant Acceleration

Motion With Constant Acceleration

In this section, we are going to derive relations among displacement, velocity, acceleration and time.
We know that, derivations
dv = adt

On integrating both sides, we get
òdv = òadt
v = at + c1 where c1 = constant of integration

If v0 be the initial velocity of the particle then at t = 0, v = v0. This implies that c1 = v0
v = v0 + at … (1)
Also we know that
derivation formula
or dx = (v0 + at)dt

On integrating, we get
òdx = òv0dt + òatdt
x = v0t + integratingat2 + c2 where c2 = constant

If x0 be the initial position of the particle, then at t = 0; x = x0. This implies that c2 = x0
x = x0 + v0t + integrating at2 … (2)

Eliminating t from equations (1) and (2), we get
v2 = quadratic equation … (3)

The equation (2) may be rewritten as

From equation (1), we have
thus equations of kinematics … (4)

The equations (1) to (4) are called the equations of kinematics in the direction along the x-axis.
The equations of kinematics are summarized as
v = v0 + at (1)
x = x0 + v0t + equation(2)
v2 = equation (3)
x = x0 + equation(v0 + v)t (4)

where x0 = initial position coordinate; x = Final position coordinate
v = final velocity; a = acceleration (constant)

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