Jane, looking for tarzan, is running at top speed (5.2 m/s) and grabs a vine hanging 4.5 m vertically from a tall tree in the jungle. how high can she swing upward?

While Jane is running has a kinetic energy, which is Ek = 1/2*m*v^2 where m is mass and v is velocity  When she grabs a vine, she is going to change the kinetic energy to potential energy.  
We know that potential energy is given by Ep = m*g*h where m is mass, g is gravity constant and h is height  
So while running the kinetic energy is Ek = 1/2 * m * 5.2^2 = 13.52*m 
Then all that energy is used to swing upward and gain potential energy 
Ep = m*g*h = Ek = 13.52*m  
m*9.8*h = 13.52*m 
h = 13.52/9.8 = 1.38 meters  
So Jane will swing 1.38 meters upward