Question

How mass and velocity affect the kinetic energy?

Answer 1

Hint: Kinetic energy of any particle or object can be defined as the energy of an object or particle due to its motion or the energy is obtained by an object from its state of rest to motion by some external force in order to answer how the mass and the velocity affect the kinetic energy we will use the formula kinetic energy = $\dfrac{1}{2}m{{v}^{2}}$
Where m is mass of object and v is the velocity of object or particle.

Formula used:
Kinetic energy (K.E) = $\dfrac{1}{2}m{{v}^{2}}$
Equation of the motion = ${{v}^{2}}-{{u}^{2}}=2as$

Complete step-by-step answer:
$\to $In order to know how the kinetic energy is affected by mass and velocity of an object or particle we will use the below stated equation.
$K.E=\dfrac{1}{2}m{{v}^{2}}$
 m is mass of object and v is the velocity of object
$\to $From the above equation we can clearly say that kinetic energy is directly proportional to mass at an object and square of velocity.
$\to $Hence when object is in motion the greater mass it bus the greater kinetic energy will become hence if you double the mass then kinetic energy will also increases similarly if the faster object is the more kinetic energy it will have if you double velocity of an object then kinetic energy will get four times greater than initial.
 $\to $ For example a rolling ball will have more kinetic energy than a cricket ball when the velocity is the same due to difference in mass.
  $\to $Similarly a car having speed of 40km/h will have more kinetic energy then the car that is having speed of 20km/h if both cars have the same mass.
$\to $Now let’s derive formula for the kinetic energy
$\to $To get an object in motion we have to apply some work on it so work done on the object.
$\begin{align}
  & \overrightarrow{W}=F.d \\
 & \overrightarrow{W}=m\times a\times d\text{ }\left( \because F=m\times a \right)......(1) \\
\end{align}$
Now we know that equation of the motion
$\dfrac{{{v}^{2}}-{{u}^{2}}}{2d}=a....\left( 2 \right)$
v = final velocity of the object
u = initial velocity of an object
a = acceleration of an object
S = distance travelled by an object
Substitute equation (2) in equation (1)
$\begin{align}
  & \overrightarrow{W}=m\times \dfrac{{{v}^{2}}-{{u}^{2}}}{2d}\times d \\
 & \overrightarrow{W}=\dfrac{1}{2}m{{v}^{2}}-\dfrac{1}{2}m{{u}^{2}} \\
\end{align}$
So basic equation for kinetic energy is
$K.E=\dfrac{1}{2}m{{v}^{2}}$
S.I unit of the kinetic energy is $kg.{{m}^{2}}.{{s}^{-2}}$

Note: Kinetic energy is scalar quantity because the dot product of the vector quantity is scalar hence the square of velocity in the equation of kinetic energy is scalar. This in turn means that kinetic energy does not have a direction associated with it.