power

The learners will calculate work done by an applied force and the power generated in practical situations such as engineering and sports science.

Definition: Power measures how fast work is done or energy is transferred in a certain time.

Main Formula: Power can be calculated using the formula:
$P = \frac{W}{t}$
where,
- P = power (in watts)
- W = amount of work done (in Joules) or energy expended (in Joules)
- t = amount of time (in seconds)
For Instantaneous Power: Power can also be found by multiplying force by velocity:
$P = F \times v$

Rate Definition:
- Power is the rate at which work is done or energy is transferred in a unit of time.
Impact of Speed on Power:
- Power is increased if work is done faster or energy is transferred in less time.
Common Mistake:
- Some people confuse energy with power. Energy is a measurement of work done, while power is how energy changes over time.

In calculus terms, power is the derivative of work with respect to time.
Relation with Work Pace:
- If work is done faster, power is higher. If work is done slower, power is smaller.

Since work is defined as force times displacement:
$W = F \times d$
And velocity is defined as displacement over time:
$v = \frac{d}{t}$
Combining these, power can be expressed:
$P = F \times v$
More power is evident when the system is both strong in force and fast in velocity.

Horsepower is often used to describe the power delivered by a machine or engine.
Definition:
- It is the power required to lift 550 pounds by one foot in one second and is approximately equivalent to 746 watts.
- 1 Hp = 550 ft-lb/s = 746 watts
- 1 hp = 745.7 watts (calculated value).

The watt is commonly associated with light bulbs.
Definition of Wattage:
- In this context, it represents the rate at which the bulb converts electrical energy into light and heat.
- A bulb with a higher wattage will use more electricity per unit of time.

If power is known, the amount of work produced can be calculated using:
$W = P \times t$
Example Calculation:
- If a bulb has a power rating of 50 watts, it will produce 50 joules per second. In an hour (3600 seconds), it will produce 180,000 joules.

Example of Walking vs. Running:
- When you walk a mile, your motive force is displacing your body, measured as work done.
- Running the same mile involves the same amount of work but completed in less time, hence the runner has a higher power rating than the walker, producing more watts.
Example with Cars:
- A car with 80 horsepower can achieve faster acceleration compared to a car with 40 horsepower; both may reach 60 miles per hour, but the former does so quicker.
Tortoise and Hare Analogy:
- In the race between a tortoise and a hare, the hare had more power and accelerated faster, but the tortoise did the same work over a longer time, indicating lower power.

Most discussions about power refer to average power, denoted as Pavg.
Definition:
- Average power is calculated as the amount of work done in a period of time:
  $P_{avg} = \frac{\Delta W}{\Delta t}$
Alternatively, it can be expressed as the amount of energy transferred in a specific timeframe:
$P_{avg} = \frac{\Delta E}{\Delta t}$

Sample Problem #1:
- A person weighing 600 N gets on an elevator. The elevator lifts the person 6 m in 10 seconds. Calculate his rate in horsepower unit.
Sample Problem #2:
- How much time is required to produce 720 Joules of work if 90 watts of power is used?
Sample Problem #3:
- A set of pulleys lifts an 800 N weight, 4 meters in 7 seconds. What power was utilized?
Sample Problem #4:
- During the PE activity, Alex runs up the stairs in 3 minutes, elevating his 80 kg body a vertical distance of 2.3 m. Determine the power generated by Alex in climbing the staircase.