Sure! Let's solve this problem step-by-step. We're asked to find the average power supplied by a 50.0 kg person who runs up a flight of stairs rising vertically by 4.0 meters in 4.6 seconds.
### Step 1: Understand the problem
To find the average power, we need to know the total work done and the time over which this work is done. The work done is the energy required to lift the person vertically by 4.0 meters.
### Step 2: Calculate the work done
Work, in this context, can be calculated as the change in gravitational potential energy (GPE). The formula for gravitational potential energy is:
[tex]\[ \text{GPE} = m \cdot g \cdot h \][/tex]
where:
- [tex]\( m \)[/tex] is the mass (50.0 kg)
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.81 m/s[tex]\(^2\)[/tex])
- [tex]\( h \)[/tex] is the height (4.0 m)
Let's plug in the values:
[tex]\[ \text{GPE} = 50.0 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 4.0 \, \text{m} \][/tex]
[tex]\[ \text{GPE} = 50.0 \times 9.81 \times 4.0 \][/tex]
[tex]\[ \text{GPE} = 1962 \, \text{J} \][/tex]
So the work done to lift the person is 1962 Joules.
### Step 3: Calculate the average power
Power is the rate at which work is done, and it can be calculated using the formula:
[tex]\[ \text{Power} = \frac{\text{Work Done}}{\text{Time}} \][/tex]
Given that the work done is 1962 Joules and the time taken is 4.6 seconds, we can calculate the average power:
[tex]\[ \text{Power} = \frac{1962 \, \text{J}}{4.6 \, \text{s}} \][/tex]
### Step 4: Solve for power
[tex]\[ \text{Power} = \frac{1962}{4.6} \][/tex]
[tex]\[ \text{Power} \approx 426.52 \, \text{W} \][/tex]
### Step 5: State the final answer
The average power supplied by the person is approximately 426.52 Watts.
