His initial energy is 0, as we assume that by Spiderman standing you mean he is at rest.
His final energy can be found by first finding the force of Spiderman swinging to the ground, which is calculated via this formula:
#F=ma#, where #F# equals force in newtons, #m# equals mass in kilograms, and #a# equals acceleration in m#/s^2#. Lets assume Spiderman's mass is #m=70kg#, and the acceleration is due to gravity, so #a=9.81m##/s^2#. Plug these values into the equation to get
#F=70*9.81#
#F=686.7# Newtons.
Now, to find work done and thus energy, we use the formula
#W=Fd#, where #W# is the change in energy measured in joules, #F# is force measured in newtons, and #d# is the distance in metres. We know that #F=686.7#, and after a bit of calculation we realise that #d=18-3=15m#. Plug these into the formula and we get
#W=686.7*15#
#W=10300.5#
#W=1.03005*10^4J#, Spiderman's final energy and so the answer to #(b)#.
To calculate Spiderman's final speed, first lets ignore any force he gains from his webs, and assume his acceleration is equal to Earth's gravity, so #a=9.81m##/s^2#. Now, to find acceleration we would normally use this formula:
#a=(u-v)/t#, where #a# is acceleration in m/#s^2#, #u# is the initial speed in m/s, #v# is the final speed in m/s and #t# is time in seconds. We know the values of #a=9.81# and #u=#(since Spiderman is starting from rest), so first we need to find #t#.
This can be done using the formula
#t=sqrt(2d/g)#, where #t# is the time in seconds, #d# is the distance in metres, and #g# is gravity in m/#s^2#. Since #d=15# and #g=9.81#, we plug them in to get
#t=sqrt(30/9.81)#
#t=sqrt(1000/327)#. We leave #t# in its exact form for now.
We now know #a#, #u# and #t#. Plug them into the previous formula to get
#9.81= (0-v)/sqrt(1000/327)#
#9.81(sqrt(1000/327))=-v#
#17.155~~-v#
#-17.155~~v#. Therefore Spiderman's final speed when his feet touch the building is roughly -17.155m/s. We have a negative because he is falling backwards towards the Earth, otherwise his magnitude is 17.155m/s.
I hope that helped!
