Question

Whats the relationship between Power and velocity

Answer 1

you might be able to work it out from this:
power = force * velocity

Answer 2

work=force X displasment
W=F*Δx
W/Δt=F*Δχ/Δt
P=F*v

Answer 3

The basic definition for power is that it is the rate at which work is done:
P = W/t
Note that the power unit is the Watt, which is defined to be equivalent to 1 Joule/second.

The standard formula for Work, W, in a case where a box of Physics books is slid at constant velocity for a distance, x, along a horizontal floor by a force, F, is
W = F*x
In this case, the energy that produced this work is expended to overcome some other force(s), often just friction. In that case, you could substitute into the power formula and obtain
P = W/t = F*x/t = F*v

Now assume a similar case without friction or any force, other than F, acting on the box. The box would accelerate. Newton's Law says that
F = m*a so that
W = F*x = m*a*x
The acceleration could be found using
a = (Vf-Vo)/t
and distance could be found using
x = (Vf+Vo)*t/2
Substituting those into P = m*a*x
W = m * (Vf-Vo)/t * (Vf+Vo)*t/2 = (1/2)*m*Vf^2 - (1/2)*m*Vo^2
So we see that in this case the energy that produced this work provides an increase in kinetic energy. The power would be considered to be the rate of increasing kinetic energy. So does the expression obtained above,
P = F*v
apply only when velocity is constant? Yea, pretty much. You could expect that some rule should have made it obvious that the velocity would be required to be constant in obtaining the
P = F*v
expression.

Answer 4

You can state next to the formula that it is valid for instantaneous Power and instantaneous velocity
P = F * dx/dt = F * v