When Emmett is lifting a box
vertically, the forces that must be added to calculate the total force are: the
gravitational force, tension force(the force exerted by Emmett to the box and
the force exerted by the box to Emmett), and air resistance force.
What question are you asking?
Mechanical energy (ME) is the sum of potential energy (PE) and kinetic energy (KE). When the toy falls, energy is converted from PE to KE, but by conservation of energy, ME (and therefore PE+KE) will remain the same.
Therefore, ME at 0.500 m is the same as ME at 0.830 m (the starting point). It's easier to calculate ME at the starting point because its just PE we need to worry about (but if we wanted to we could calculate the instantaneous PE and KE at 0.500 m too and add them to get the same answer).
At the start:
ME = PE = mgh
ME = 0.900 (9.8) (0.830)
ME = 7.32 J
The intensity of the magnetic force F experienced by a charge q moving with speed v in a magnetic field of intensity B is equal to

where

is the angle between the directions of v and B.
1) Re-arranging the previous formula, we can calculate the value of the magnetic field intensity. The charge is

. In this case, v and B are perpendicular, so

, therefore we have:

2) In this second case, the angle between v and B is

. The charge is now

, and the magnetic field is the one we found in the previous part, B=2.8 T, so we can find the intensity of the force experienced by this second charge: