Question

Define power. Prove that $P=F.v$, where the symbols have their usual meanings.

Answer 1

Hint: Power is defined as the amount of energy transferred per unit time. Here, we will use the formula of power, that is, the ratio of energy used to the time taken. Here, we will use the formula of work done in place of energy and we will put this formula in the formula of power.

Formula used:
The formula used for calculating the power is given below
$P={\displaystyle \frac{E}{t}}$
Here, $P$ is the power, $E$ is the amount of energy and $t$ is the time taken.
The formula of energy will be in the form of work done and is given below
$E=W=F\times \mathrm{\Delta }x$
Here, $W$ is the work done, $F$ is the force required for the work and $\mathrm{\Delta }x$ is the displacement.

Complete step by step answer:
Power is defined as the amount of energy transferred per unit time. It is also defined as the rate of doing work or activity in the minimum time.
Here, the formula used for calculating the power is given below
$P={\displaystyle \frac{E}{t}}$
Here, the energy will be equal to the work done that will be equal to the product of force and displacement and is given below
$E=W=F\times \mathrm{\Delta }x$
Putting, this value in the formula of power as shown below
$P={\displaystyle \frac{F\times \mathrm{\Delta }x}{t}}$
$\Rightarrow P=F\times {\displaystyle \frac{\mathrm{\Delta }x}{t}}$
Here, ${\displaystyle \frac{\mathrm{\Delta }x}{t}}$ is known as the rate of change of displacement with respect to time. This is known as velocity and is denoted by $v$.
$\therefore v={\displaystyle \frac{\mathrm{\Delta }x}{t}}$
Putting this value of velocity in the in the above equation, we get
$P=F\times v$
Hence, we get $power=force.velocity$

Hence, we have proved that $P=F.v$.

Note:We know that the power has only one direction and the measurement of power is one dimensional. Therefore, we can say that the power is a scalar quantity. This is because when we have the dot product of something, the quantity will be scalar. Also, the power is the product of force and the volume.