A 55.0 kg girl is roping through the top of a tiny tree. She quickly swings down from her 12.0 m high perch and grabs a 21.0 kg dog as she swings. They barely swing back up to a tree limb. How high is this tree limb?

you know the initialPE, and the final PE is not known

intial=final
55*g*12=(55+21)g*h
solve for h.

To determine the height of the tree limb, we can make use of the principle of conservation of energy. The total mechanical energy at the top of the swing (initial point) will be equal to the total mechanical energy at the tree limb (final point).

At the top of the swing, the only form of energy the girl-dog system has is potential energy, given by:

PE = m * g * h

Where:
m = mass of the system (girl + dog) = 55.0 kg + 21.0 kg = 76.0 kg (total mass)
g = acceleration due to gravity = 9.8 m/s^2
h = height = 12.0 m

So, the initial potential energy (PE_initial) is:

PE_initial = 76.0 kg * 9.8 m/s^2 * 12.0 m

Now, at the tree limb, the system has both potential energy (due to the height) and kinetic energy (due to the motion). The total mechanical energy is the sum of these two energies:

PE_final + KE_final = PE_initial

Since the system comes to rest at the highest point, the final kinetic energy (KE_final) is zero. Therefore, we have:

PE_final = PE_initial

To find the final potential energy (PE_final), we rearrange the equation:

PE_final = PE_initial - KE_final

Since KE_final = 0, the equation simplifies to:

PE_final = PE_initial

Substituting the values:

PE_final = 76.0 kg * 9.8 m/s^2 * 12.0 m

Calculating this, we find:

PE_final = 8,902.4 joules

So, the height of the tree limb is approximately 8,902.4 joules.