.A particle slows down from a speed v0 while moving in a straight line. If declaration is direct proportional to the particle its displacement as a function of time is ( K is proportionality constant )
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Explanation:
The average velocity is given as:
v
av
=
t
s
=
Totaltime
Totaldisplacement
We know a = v
ds
dv
⇒ v dv = a ds
Let us calculate the displacement by substituting a = −α
v
(for retardation) in vdv = a ds.
We have v dv = (−α
v
ds ⇒
v
dv = −αds
When the particle slows down from v
0
to 0, it covers a distance s.
Hence, ∫ v in0
0
v
dv=−∫
0
s
αds
This gives s =
3α
2v
0
3/2
Now we will find the time of motion by substituting a = −α
v
in a =
dt
dv
, we get
−α
v
=
dt
dv
⇒
v
dv
=−αdt
If the particle takes time t to stop, we have
∫
v
0
0
v
dv
=−α∫
0
t
dt
This yields t =
α
v
0
Finally, substituting s and t in v
av
= s/t, we have
v
av
=
3
2v
0
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