How can the alternative definition of power:

P = \frac{w}{\Delta t} = F \frac{d}{\Delta t}

... can be derived by substituting the definitions of work and speed into the standard definition of power, P =\frac{W}{\Delta t}.

(Equations written without latex formatting:)
P = w/delta t = F(d/delta t)

P = W/delta t

I don't understand what the question wants, so I can't show why I have done so far sorry

We do not support LaTeX here, but you can use / for division, and unicode symbols for special characters.

Power = work / time
work = Force * distance
speed = distance / time
force * speed = force * distance / time

I believe the four equivalences are sufficient for you to get started.

No problem! I can explain the process to you step by step.

The standard definition of power is given by:

P = W / Δt

Where P is power, W is work, and Δt is the time interval during which the work is done.

To derive the alternative definition of power, we can start by substituting the definitions of work and speed into the standard power equation.

The definition of work is given by:

W = F * d

Where W is work, F is the applied force, and d is the displacement of the object in the direction of the force.

Now, let's substitute this definition into the standard power equation:

P = W / Δt
P = (F * d) / Δt

Since speed (v) is defined as the displacement divided by the time interval:

v = d / Δt

We can rewrite the equation as:

P = F * v

This is the alternative definition of power, where power (P) is equal to the force (F) applied to the object multiplied by the speed (v) at which the object is moving.

In summary, by substituting the definitions of work and speed into the standard power equation, we can derive the alternative definition of power as P = F * v.

To derive the alternative definition of power, we first substitute the definitions of work and speed into the standard definition of power, P = W / Δt.

1. Starting with the standard definition of power: P = W / Δt

2. Substituting the definition of work into the equation:
P = (F * d) / Δt

Here, F represents the force applied and d represents the displacement.

3. We can rearrange the equation to separate the force and displacement terms:
P = (F * d) / (Δt * 1)

Since speed is defined as the displacement divided by the time taken, we can replace the term d / Δt with speed (v):
P = (F * v) / 1

4. Finally, simplifying the expression gives us the alternative definition of power:
P = F * v

This equation shows that power can also be calculated by multiplying the force applied by the object's speed or velocity.