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# 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.

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,

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

uvdv = ∫0t adt = a∫0tdt =

[v]uv = a[t]0t

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,

0sds = ∫0tudt+∫0tatdt

S=ut+1/2at2

(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,

0sds =∫uv(vdv)/a

S=1/2a(v2-u2)

Or

2as=v2-u2

v2=u2+2as

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

Expression for the distance travelled in nth 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,

Sn = distance travelled during first n seconds ? distance

travelled during (n ?1) seconds

Distance travelled during n seconds

Dn=un+1/2an2

Distance travelled during (n -1) seconds

D(n-1) = u(n+1) + 1/2a(n-1)2

the distance travelled in the nth second = Dn− D(n ?1)

Sn = (un+1/2an2).[u(n-1)+1/2a(n-1)2]

Sn=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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11th 12th std standard Class Physics sciense Higher secondary school College Notes : Equations of motion |