If we define the initial velocity as having magnitude #u#, then the vertical component of this is #u*sin30# (the horizontal component would be #u*cos30# but we are not interested in this as it has no effect on potential energy)
A word of caution: there is no definition or reference in the question as to what 30 degrees is measured from; it could be from the horizontal or the vertical. | have assumed from the horizontal. Please let me know if this is not the case and I will re-work.
The body will travel a maximum distance #s# upwards. At this point its vertical velocity will be nil. We can used the following to determine #s#:
#v^2=u^2+2as#
We know #v#, final velocity is nil, and #u#, initial velocity is #usin30# and #a# will be acceleration due to gravity, #g# (which will be negative relative to the direction of the initial velocity). Hence:
#0=u^2sin^2 30-2gs#
#2gs=u^2sin^2 30#
#s=(u^2sin^2 30)/(2g)#
So final potential energy will be:
#PE=mgh=mgs#
#PE=(mg)*(u^2sin^2 30)/(2g)#
#PE=(m*u^2sin^2 30)/(2)#
Initial kinetic energy is
#KE=1/2m*u^2#
So percentage conversion is:
#"PE"/"KE"=(m*u^2sin^2 30)/(2)/(1/2m*u^2)#
#"PE"/"KE"=sin^2 30=0.25#
So the answer is 25%
