Question

A child of mass \(15 \mathrm{kg}\) climbs to the top of a slide that is $1.7 \mathrm{m}\( above a horizontal run that extends for \)0.50 \mathrm{m}$ at the base of the slide. After sliding down, the child comes to rest just before reaching the very end of the horizontal portion of the slide. (a) How much internal energy was generated during this process? (b) Where did the generated energy go? (To the slide, to the child, to the air, or to all three?)

Short answer

Question: Calculate the internal energy generated during the sliding process and determine where the generated energy went. Answer: The internal energy generated during the sliding process is W_internal = E_mech, initial - E_mech, final. In this case, W_internal = (15 kg)(9.81 m/s^2)(1.7 m) - 0 = 249.735 J (approximately). The generated energy goes into heating up the slide, the child's body, and the air, as well as any sound produced during the slide.

Step-by-step solution

(a) Calculate the initial potential energy

The initial potential energy, \(U_i\), possessed by the child at the top of the slide can be calculated using the formula \(U_i = mgh\) where \(m\) is the mass of the child, \(g\) is the acceleration due to gravity, and \(h\) is the height of the slide. In this case, \(m = 15 \mathrm{kg}\), \(g = 9.81 \mathrm{m/s^2}\), and \(h = 1.7 \mathrm{m}\). Therefore, \(U_i = (15 \mathrm{kg})(9.81 \mathrm{m/s^2})(1.7 \mathrm{m})\).

Calculate the final gravitational potential energy

The child comes to rest at the very end of the horizontal portion of the slide. At this point, all gravitational potential energy has been transferred to other forms of energy, so the final gravitational potential energy, \(U_f\), is 0.

Calculate the total mechanical energy at the beginning and end

The child's initial kinetic energy is also 0 since they start at rest. The total mechanical energy, \(E_{mech}\), at the beginning, is equal to the initial potential energy, \(U_i\). At the end, the total mechanical energy is the sum of final kinetic energy, \(K_f\), and final potential energy, \(U_f\). Since both \(K_f\) and \(U_f\) are 0, the total mechanical energy at the end is also 0.

Apply conservation of mechanical energy

Conservation of mechanical energy states that the total mechanical energy at the beginning and end of a process is equal if no external forces do work on the object. In this case, the total mechanical energy is not conserved since friction forces are doing work. The change in total mechanical energy is equal to the work done by external forces, which is the negative of the internal energy generated: \(ΔE_{mech} = E_{mech, final} - E_{mech, initial} = -W_{internal}\).

Calculate the internal energy generated

Rearranging the above equation, we can find the internal energy generated as \(W_{internal} = E_{mech, initial} - E_{mech, final}\). We previously calculated \(E_{mech, initial}\) and \(E_{mech, final}\) and know that \(E_{mech, initial}\) is not equal to \(E_{mech, final}\). Therefore, substitute the calculated values to find the internal energy generated.

(b) Determine where the energy went

The internal energy generated is due to the action of external forces, such as friction between the slide and the child, and air resistance as the child slides down. So the generated energy goes into heating up the slide, the child's body, and the air, as well as any sound produced during the slide. Combining these steps, we can find the internal energy generated during the sliding process and determine where the generated energy went.