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.
∫uvdv = ∫0t adt = a∫0tdt =
[v]uv = a[t]0t
(ii) The velocity of the body is given by the first derivative of the displacement with respect to time.
Since v = u + at, ds = (u + at) dt
The distance s covered in time t is,
∫0sds = ∫0tudt+∫0tatdt
(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
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
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]
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.