Mechanical Power
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P = W/t measures average power as work done divided by the time it took, while P = Fv gives instantaneous power as the product of force and velocity at a given moment. They describe the same physical quantity, but the first suits a completed task over an interval and the second suits a snapshot of an ongoing process.

Power measures the rate at which work is done, not the total work itself, so a motor that completes a task in half the time delivers twice the power of one that takes twice as long for the identical task. This is precisely why power, not total work, is the number quoted on engines, motors, and appliances — it tells you how fast they can deliver that work.

No — energy (or work) is measured in joules and represents a total quantity, while power is measured in watts and represents a rate, joules per second. Confusing the two is common in everyday language ('using a lot of power') when what's often meant is total energy consumed over time.

The work done against gravity (mgh) is identical either way since it depends only on mass, gravity, and height gained, not on speed — but power is that work divided by time, so completing the climb faster means dividing the same work by a smaller time, yielding higher power. This is a common example of why 'more power' doesn't necessarily mean 'more total energy.'

Horsepower is a non-SI power unit, historically defined to compare steam engines to draft horses, and one mechanical horsepower equals approximately 746 watts. If your result is in watts and you need horsepower (or vice versa), that conversion factor is the bridge between the two unit systems.