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Question

Which of the following relations are always true?
v=velocity, a=acceleration, K=12mv2= kinetic energy:

A
dKdt=mv.a
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B
d|v|dt=a.v|v|
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C
d|v|dt=dadt
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D
Δ|a|=t2t1a dt
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Solution

The correct option is B d|v|dt=a.v|v|
For option (A)
K=12mv2

dKdt=12m(2v)dvdt
=mva=m(v.a)
Hence, this option is correct.

For option (B)

dvdt=a.vv=avcos(av)v [cancle v both above and below]
so, this option is also true because dvdt means change in magnitude of velocity which is that acceleration which is in the direction of velocity and dvdt also meant by acceleration
Which is in the direction of velocity.

For option (C)

dvdt∣ ∣dadt∣ ∣
This option is false because in magnitude of velocity may be negative because of retardation.

For option (D)

Δ|a|=t2t1a dt
this option is false because L.H.S is meant for change in magnitude of acceleration and R.H.S meant for area of at graph

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