a person's metabolic processes can usually operate at a power of 6 W/kg of body mass for several hours at a time. If a 60 kg woman carrying a 12 kg pack is walking uphill with an energy-conversion efficiency of 20 percent, at what rate, in meters/hour, does her altitude increase?
367 m/h
56N
35N
To determine the rate at which her altitude increases, we first need to calculate the amount of power the woman is generating while walking uphill. Here's how we can approach this step-by-step:
Step 1: Calculate the total mass
The total mass is the sum of the woman's body weight and the weight of the pack.
Total mass = woman's body mass + pack mass
Total mass = 60 kg + 12 kg
Total mass = 72 kg
Step 2: Calculate the metabolic power
Given that a person's metabolic processes can operate at a power of 6 W/kg of body mass, we can calculate the metabolic power generated by the woman.
Metabolic power = Power per unit mass * Total mass
Metabolic power = 6 W/kg * 72 kg
Metabolic power = 432 W
Step 3: Calculate the useful power
Since the energy-conversion efficiency is given as 20 percent, we need to calculate the useful power, which is a fraction of the metabolic power.
Useful power = Energy conversion efficiency * Metabolic power
Useful power = 0.20 * 432 W
Useful power = 86.4 W
Step 4: Calculate the work done against gravity
The work done against gravity is the useful power generated while walking uphill, as it accounts for the increase in altitude.
Work done = Useful power
Step 5: Calculate the rate of altitude increase
The rate of altitude increase can be determined by dividing the work done against gravity by the force exerted against gravity, which is equal to the weight of the woman and the pack.
Rate of altitude increase = Work done / Force against gravity
Before proceeding, we need the value of gravitational acceleration (g). Assuming a standard value, g = 9.8 m/s².
Now, let's calculate the rate of altitude increase:
Rate of altitude increase = Work done / (Mass * g)
Substituting the values:
Rate of altitude increase = 86.4 W / (72 kg * 9.8 m/s²)
Rate of altitude increase ≈ 0.122 m/s
However, since we need the rate in meters/hour, we'll convert it:
Rate of altitude increase ≈ 0.122 m/s * 60 * 60
Rate of altitude increase ≈ 439.2 m/h
Therefore, the rate at which her altitude increases while walking uphill is approximately 439.2 meters per hour.