Question

If a force F is applied on a body and it moves with a velocity $v$, the power will be:
A) $F \times v$
B) $\dfrac{F}{v}$
C) $\dfrac{F}{{{v^2}}}$
D) $F \times {v^2}$

Answer 1

Know that Power is the capacity to change or influence the behaviour or course of events of an object by supplying mechanical or electrical energy. Here, we have to express power in terms of force to get the formula.

Complete step by step solution:
Power has many definitions and can be expressed in various forms. From these definitions, we can find the required expression.
Here, the question states a force $F$ being applied on a body to make it move with velocity $v$ . This means that the force here is used to do mechanical work of moving the body. Hence, the power here will be mechanical power. Power can be mechanical power or electrical power.

Mechanical power is defined as the rate of work done by an object. Also defined as transfer of energy or conversion of energy per unit time.
$Power = \dfrac{{Work}}{{time}}$
Also expressed as
$Power = \dfrac{{\Delta Energy}}{{\Delta Time}}$
SI unit of power is $Watt$ or $Joule/s$ also written as $kg{m^2}{s^{ - 3}}$

Whenever we apply force on an object, the object gains some energy. The energy can be in the form of potential energy or kinetic energy. So, we can express energy in terms of force.
Mathematically, energy is equal to the product of the applied force and displacement along the direction of force.
$\Rightarrow E = F \cdot d$
$\Rightarrow E = Fd\cos \theta$
Where $E$ is the energy
$F$ is force
$d$ is displacement
$\theta$ is the angle between the applied force and the displacement

We now substitute this equation of energy in the equation of power and get
$P = \dfrac{{F \cdot d}}{t}$
But we also know that displacement of the object per unit time is known as velocity of the object.
$v = \dfrac{d}{t}$
Where $v$ is velocity of the object
$d$ is the displacement
$t$ is time
Substituting this, we get:
$Power = Force \times velocity$
$\Rightarrow P = F \times v$

Therefore, option $(A), F \times v$ is the correct option.

Note: Power is equal to the dot product of the force and velocity. In this question $F \times v$ solely means the product of force and velocity and does not mean the cross product between them. Power can be positive or negative with increase or decrease with energy per unit time, but energy can never be negative as an object always has some energy. Energy can only increase or decrease but always remains positive.
Electric power is expressed as $Power = voltage \times current$