In mechanics, we can also express power in terms of force and velocity. In this post, we will derive the

relation between power and velocity. First, we will follow a simpler method (for class 10). And then we will use the concept of Vector to find out theForce power velocity relationship.

**Relation between power and velocity**

Consider a vehicle traveling at constant velocity v along a straight, level road. The engine must continue to do work against friction as the vehicle is moving at a constant velocity.

If the frictional force is F, then the engine will supply an equal-sized force in the opposite direction.

The work done by the engine, W, in time t, is Fs, where s is the distance traveled in time t:

So, power = (Fs)/t ………………. (1)

but s/t= v,

therefore: power = Fv

Now, let’s find the above relationship involving vector concepts.

**Force power velocity relationship** [using vector concept]

Suppose that a force ** F** acts on an object while it undergoes a vector displacement

**Δ**. If

*s*

*F*_{||}is the component of

**tangent to the path (parallel to**

*F***Δ**

*), then the work done by the force is ΔW =*

**s***F*

_{||}Δs.

The average power is P_{av} =(*F*_{||}Δs)/Δt = *F*_{||} (Δs/Δt) = *F*_{||}*v*_{av} ……………… (a)

Instantaneous power P is the limit of this expression as Δt ->0

P = *F*_{||}*v* ………………(b)

where v is the magnitude of the instantaneous velocity.

We can also express Eq. (b) in terms of the scalar product: P = F.v

**Numerical solved using P= Fv equation**

A cyclist is travelling along a straight level road at a constant velocity of 27 km h

^{−1}against total frictional forces of 50 N.

Calculate the power developed by the cyclist.

**Solution:**

First, let’s Convert the velocity from km h^{−1} into m s^{−1}:

v= 27 km h^{−1} = 27 ×1000/3600 = 7.5 m s−1.

As the cyclist is moving with uniform velocity, hence Force applied by the cyclist = frictional force

So the power developed by the cyclist = force × velocity = 50 x 7.5 W =375 W