The superheroine Xanaxa, who has a mass of 67.5 kg , is pursuing the 70.3 kg archvillain Lexlax. She leaps from the ground to the top of a 179 m high building then dives off it and comes to rest at the bottom of a 15.9 m deep excavation where she finds Lexlax and neutralizes him. Does all this bring about a net gain or a net loss of gravitational potential energy?

Where y1 is the initial position where the energy is defined as zero, in general in problems on the Earth's surface the reference system is taken at the lowest point, whereby y1 = 0 and consequently U1 = 0, this It is a matter of comfort since what matters are the changes in energy and not its value at one point explained.

Let's apply these concepts to our problem.

Jump from the ground to 179 m, let's calculate the energy change

ΔU1 = mg (y2-y1)

ΔU1 = 67.5 9.8 (179 -0)

ΔU1 = 1.18 10⁵ J gains energy

Dives up to Y3 = -15.9m below ground, therefore, this height is negative

The total height is y = 179 - (-15.9) = 194.9 m

ΔU2 = mg (y3-y2)

These are the points where the dive begins and where it arrives

ΔDU2 = 67.5 9.8 (- 194.9)

ΔU2 = -1.29 105 J loses energy

The total change in potential energy is

Ut = ΔU1 + ΔU2

Ut = 1.18 10⁵ - 1.29 10⁵

Ut = -0.11 10⁵ J

Therefore, there is a net loss of potential energy

You can also calculate ΔU = mg (Δy)

Where ΔY is the change in body position, that is -15.9 m

ΔU = 67.5 9.8 (-15.9)

ΔU = -1.1 10⁴ J = -0.11 10⁵ J

You can see that it is the same value.