We can calculate the average mechanical power generated by the mountain climber using the formula:

$P = \frac{W}{t}$

Where:

• $P$ is the power in watts (W),
• $W$ is the work done in joules (J),
• $t$ is the time taken in seconds (s).

First, we calculate the work $W$, which is the gravitational potential energy gained by the climber when climbing the hill:

$W = mgh$

Where:

• $m = 60.0 \, \text{kg}$ (mass of the climber),
• $g = 9.81 \, \text{m/s}^2$ (acceleration due to gravity),
• $h = 405 \, \text{m}$ (height of the hill).

Step 1: Calculate Work (W)

$W = 60.0 \times 9.81 \times 405$ $W = 238,803 \, \text{J}$

Step 2: Convert Time to Seconds

The time given is $40.0 \, \text{min}$. Convert this to seconds: $t = 40.0 \, \text{min} \times 60 \, \text{s/min} = 2400 \, \text{s}$

Step 3: Calculate Power (P)

Now, calculate the average power: $P = \frac{238,803}{2400} = 99.5 \, \text{W}$

Thus, the average mechanical power that the mountain climber must generate is approximately 99.5 W.

Would you like to dive deeper into the energy conversion process or have any questions?

5 Related Questions:

1. How would the power change if the hill were steeper but the same height?
2. What is the effect of a higher body mass on the required mechanical power?
3. How does the efficiency of converting chemical to mechanical energy affect the actual power output?
4. What is the difference between mechanical power and metabolic power?
5. How does the altitude affect the climber’s potential energy?

Tip:

For better efficiency, climbers often pace themselves to reduce sudden energy demands, improving overall stamina.