A 34.0 kg child slides down a long slide in a playground. She starts from rest at a height h1 of 22.00 m. When she is partway down the slide, at a height h2 of 11.00 m, she is moving at a speed of 8.40 m/s. Calculate the mechanical energy lost due to friction (as heat, etc.).

Well, sliding down a slide sounds like a slippery situation! We can calculate the mechanical energy lost due to friction using the law of conservation of mechanical energy. Let's get started!

The initial potential energy of the child at height h1 is given by PE1 = mgh1, where m is the mass of the child (34.0 kg), g is the acceleration due to gravity (9.8 m/s²), and h1 is the initial height (22.00 m).

PE1 = (34.0 kg)(9.8 m/s²)(22.00 m)

The kinetic energy of the child halfway down the slide at height h2 is given by KE = (1/2)mv², where v is the velocity of the child (8.40 m/s) at that height.

KE = (1/2)(34.0 kg)(8.40 m/s)²

To find the potential energy at height h2, we can use the formula PE2 = mgh2.

PE2 = (34.0 kg)(9.8 m/s²)(11.00 m)

Now, the mechanical energy lost due to friction can be calculated as the difference between the initial and final mechanical energies:

Energy lost = PE1 + KE - PE2

I'll leave it to you to crunch the numbers and find out how much mechanical energy the child loses on her slippery slide adventure!

To calculate the mechanical energy lost due to friction, we first need to find the initial mechanical energy and final mechanical energy of the child.

The initial mechanical energy of the child is given by the potential energy at height h1:
Initial mechanical energy (Ei) = mgh1

where m = mass of the child (34.0 kg), g = acceleration due to gravity (9.8 m/s^2), and h1 = initial height (22.00 m).

So, Ei = 34.0 kg * 9.8 m/s^2 * 22.00 m = 71892 J (Joules)

The final mechanical energy of the child is the sum of the kinetic energy and potential energy at height h2:
Final mechanical energy (Ef) = (1/2)mv^2 + mgh2

where v = final velocity (8.40 m/s), and h2 = final height (11.00 m).

The kinetic energy (KE) of the child is given by (1/2)mv^2:
KE = (1/2) * 34.0 kg * (8.40 m/s)^2 = 11935.68 J

The potential energy (PE) of the child at height h2 is given by mgh2:
PE = 34.0 kg * 9.8 m/s^2 * 11.00 m = 37568 J

So, Ef = KE + PE = 11935.68 J + 37568 J = 49403.68 J (Joules)

The mechanical energy lost due to friction is the difference between the initial and final mechanical energy:
Energy lost due to friction = Ei - Ef

Energy lost due to friction = 71892 J - 49403.68 J = 22488.32 J (Joules)

Therefore, the mechanical energy lost due to friction is 22488.32 Joules.