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Chapter: 11th 12th std standard Class Physics sciense Higher secondary school College Notes

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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,

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

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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