Equations of motion

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^vdv = ∫₀^t adt = a∫₀^tdt =

[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^sds = ∫₀^tudt+∫₀^tatdt

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,

∫₀^sds =∫_u^v(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 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+1/2an²

Distance travelled during (n -1) seconds

D_(n-1) = u(n+1) + 1/2a(n-1)²

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

S_n = (un+1/2an²).[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.