0Consider the following :
A particle moves along a straight line as per law:Power is proportional to displacement raised to power -3/2. That is,P∝S-3/2.Then, acceleration is proportional to which power of velocity?
Solution:(usual method) P=Fv=mav= m(dv/dt)v=m(dv/ds)(ds/dt)v=m(dv/ds)v2
given:P∝S-3/2 ⇒(dv/ds)v2=S-3/2
Integrating,v3∝s-1/2 . v∝ s-1/6. Therefore,a∝(s-3/2)/v =(v-9)/v =v8.
Hence,Answer is , acceleration is proportional to 8th power of velocity.
An easier and general method for such problems:Let s∝tn
then,v∝tn-1 and a∝tn-2
Hence av∝s-3/2 means tn-1 tn-2 =t-3n/2 ⇒n=6/7,a∝t-8/7 and v∝t-1/7 and hence a∝v8
