For uniformly accelerated motion, some simple equations that relate displacement s, time t, initial velocity u, final velocity v and acceleration a are obtained.

*Equations of motion*

For uniformly accelerated motion, some simple
equations that relate displacement s, time *t*,
initial velocity *u*, final velocity *v* and acceleration *a* are obtained.

(i) As acceleration of the body at any instant
is given by the first derivative of the velocity with respect to time,

A=dv/dt
or dv=adt

If the velocity of the body changes from u to v
in time t then from the above equation.

∫_{u}^{v}dv = ∫_{0}^{t}
adt = a∫_{0}^{t}dt =

[v]_{u}^{v} = a[t]_{0}^{t}

v-u =at

v=u+at

(ii) The velocity of the body is given by the
first derivative of the displacement with respect to time.

v=ds/dt

ds=vdt

Since v = u + at, ds = (u + at) dt

The distance s covered in time t is,

∫_{0}^{s}ds = ∫_{0}^{t}udt+∫_{0}^{t}atdt

S=ut+1/2at^{2}

(iii) The acceleration is given by the first derivative of velocity
with respect to time. (i.e)

A=dv/dt=dv/ds.ds/dt = dv/ds.v

Ds=1/a.vdv

Therefore,

∫_{0}^{s}ds =∫_{u}^{v}(vdv)/a

S=1/2a(v^{2}-u^{2})

Or

2as=v^{2}-u^{2}

v^{2}=u^{2}+2as

The equations (1), (2) and (3) are called equations of motion.

**Expression for the distance travelled in n ^{th}
second**

Let a body move with an initial velocity u and travel along a straight
line with uniform acceleration a.

Distance travelled in the nth second of motion is,

S_{n} = distance travelled during first n seconds ?
distance

travelled during (n ?1) seconds

Distance travelled during n seconds

D_{n}=un+1/2an^{2}

Distance travelled during (n -1) seconds

D_{(n-1)} = u(n+1) + 1/2a(n-1)^{2}

the distance travelled in the nth second = D_{n}− D_{(n
?1)}

S_{n} = (un+1/2an^{2}).[u(n-1)+1/2a(n-1)^{2}]

S_{n}=u+1/2a(2n-1)

*Special Cases*

Case (i) : For downward motion

For a particle moving downwards, *a = g*, since the particle moves in the
direction of gravity.

Case (ii) : For a freely falling body

For a freely falling body, *a = g *and* u *= 0,
since it starts from rest.

Case (iii) : For upward motion

For a particle moving upwards, *a =* *-* *g,* since the particle
moves against the gravity.

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