Final answer:
To calculate the metabolic power for a 68 kg hiker walking up a 9% slope at 5.0 km/h with 25% efficiency, one must add the power required to walk on level ground (380 W) to the power needed to lift the hiker against gravity. Finally, divide the total mechanical power by the efficiency to obtain the metabolic power, which is approximately 1853.4 W.
Step-by-step explanation:
The question is about calculating the metabolic power required by a 68 kg hiker walking up a 9% slope at a speed of 5.0 km/h, with a metabolic efficiency of 25%.
Firstly, we need to find the power required to walk on level ground, which is given as 380 W. Next, we need to calculate the power needed to raise the hiker's body vertically against gravity. The slope indicates the vertical rise for every 100 horizontal meters; so, a 9% slope means a 9 m vertical rise per 100 m. To get power, we need to find work done against gravity, which is the force due to gravity times the height (W = mgh), and divide this by the time taken to ascend that height. Since power is the rate at which work is done, this gives us the mechanical power. However, due to the efficiency of 25%, we need to divide this mechanical power by the efficiency to find the metabolic power.
Calculating the metabolic power, we have:
- The hiker's mass (m) is 68 kg.
- Gravitational acceleration (g) is approximately 9.81 m/s2.
- To find the vertical speed (vy), we multiply the horizontal speed by the slope: 5.0 km/h * 9% = 0.45 km/h. We convert this to m/s (0.45 km/h = 0.125 m/s).
- Calculate the power to raise the hiker: Pgrav = mgh/t, where h/t is the vertical speed, so Pgrav = mgvy.
- Actual metabolic power: Pmetabolic = Plevel + Pgrav and Ptotal = Pmetabolic / efficiency.
Implementing the formula:
Pgrav = 68 kg * 9.81 m/s2 * 0.125 m/s = 83.35 W
The total mechanical power is 380 W + 83.35 W = 463.35 W.
The necessary metabolic power is 463.35 W / 0.25 = 1853.4 W
Therefore, the hiker needs a metabolic power of approximately 1853.4 W to walk up the 9% incline at 5.0 km/h.