A body of mass m moving with velocity u collides with another ball of mass m2 at rest. the ratio m1/m2 for maximum energy transfer is:

ans:1

in the worked out solution:

ΔK.E/K.E has been worked out as 4n/(1+n)^2

where n=m1/m2 and ΔK.E is decrease in K.E.

then, d/dn(4n/(1+n)^2) = 0

so n=1

can you please explain the differentiation part?

Is there any other method?

change of KE /KE = 4n/(n+1)² or

change in KE depend upon 4n/(n+1)²        for a particular value of initial KE...

this means dKE is a function of n                 (where n is variable...)

now we have to maximise this change for this we can use concept of maxima & minima ...

dKE = f(n) = 4n/(n+1)²

 differentiating this eq wrt n

      d/dn [f(n)] = d/dn [4n/(n+1)² ]

                     =4[-n² + 1]/(n+1)⁴                           (using quotient rule)

  now putting the above expression equal to 0

             -n²+1 = 0

                  n=+1 or -1

now we have two values for n ,on substituting these values we will see that function

 is maximum at n=1 & minimum at n=-1 so n=1 is the value for which kinetic energy transferred is maximum...