Final answer:
To determine the total power a 75 kg person must produce to run up stairs with a vertical height of 4.0 m in 5 seconds, one must calculate the potential energy (PE) for height gain, kinetic energy (KE) at the top, sum them for total work (W), calculate power output, and divide by efficiency (25%).
Step-by-step explanation:
The question asks for the total power that must be produced by a 75 kg person climbing a flight of stairs to a vertical height of 4.0 m in 5 seconds, while having 25% efficiency. To calculate the work done, we need to consider both the potential energy gained and the kinetic energy at the top. The potential energy (PE) gained can be calculated using the formula:
Where m is the mass of the person, g is the acceleration due to gravity (9.81 m/s2), and h is the vertical height.
The kinetic energy (KE) at the top can be calculated using:
Where v is the final velocity.
The total work done, W, is the sum of PE and KE:
The power output, which is work done per unit time (t), is:
Since the person is 25% efficient, the actual power produced is four times the power output:
Plugging in the values,
By calculating the values, we can find the required total power output.