Prove that the instantaneous power is equal to the dot product of force and velocity
To prove that the instantaneous power is equal to the dot product of force and velocity, we start with the definition of power. Power is given by the rate at which work is done or energy is transferred, and is given by the equation:
Power = ΔWork / ΔTime
Now, let's consider a particle moving in a straight line. The work done on the particle by a force F over a displacement Δs is defined as the dot product of the force and the displacement, which can be written as:
ΔWork = F ⋅ Δs
Now, let's consider a small time interval Δt during which the displacement is Δs. The average velocity during this time interval is given by:
Average Velocity = Δs / Δt
The instantaneous velocity at any given moment is the limit of the average velocity as the time interval approaches zero, which can be written as:
Instantaneous Velocity = lim (Δt → 0) (Δs / Δt) = ds/dt
Taking the derivative of the displacement with respect to time gives us the instantaneous velocity.
Now, let's rewrite the equation for work in terms of force and velocity:
ΔWork = F ⋅ Δs
Since Δs = (ds/dt) Δt, we can substitute this into the equation:
ΔWork = F ⋅ (ds/dt) Δt
Dividing both sides by Δt, we get:
(ΔWork / Δt) = F ⋅ (ds/dt)
But (ΔWork / Δt) is the definition of power, so we have:
Power = F ⋅ (ds/dt)
which is the dot product of the force and velocity. Hence, the instantaneous power is equal to the dot product of force and velocity, as proved.