Question

A body of mass \[1\,{\text{kg}}\] is thrown upwards with a velocity \[20\,{\text{m/s}}\].It momentarily comes to rest after attaining a height of \[18\,{\text{m}}\] .How much energy is lost due to air friction? \[\left( {g = 10\,{\text{m/}}{{\text{s}}^{ - 2}}} \right)\]

Answer 1

Hint: First of all, we will find the kinetic energy and then potential energy. We can calculate the energy lost due to air friction by simply finding the difference between the two.

Step by step answer: In the given question, we are provided the following details:
Mass of the body is \[1\,{\text{kg}}\] .
Initial velocity with which the body is thrown upwards is \[20\,{\text{m/s}}\] .
The height at which the body comes to rest is \[18\,{\text{m}}\] .
We are asked to find the energy lost due to air friction.
First, we will find the kinetic energy of the body when it was thrown up in the air.
The expression of the kinetic energy is given by:
\[KE = \dfrac{1}{2}m{v^2}\] …… (1)
Where,
\[KE\] indicates kinetic energy.
\[m\] indicates mass of the body.
\[v\] indicates velocity of the body.
Substituting the required values in the equation (1), we get:
\[
  KE = \dfrac{1}
{2}m{v^2} \\
  \Rightarrow KE = \dfrac{1}
{2} \times 1 \times {20^2} \\
 \Rightarrow KE = \dfrac{{400}}
{2}\,{\text{J}} \\
 \Rightarrow KE = 200\,{\text{J}} \\
\]
The kinetic energy associated with the body is found out to be \[200\,{\text{J}}\] .
When we throw a body, at some point of time after reaching a certain height, the kinetic energy associated with the body is lost. It is lost because the velocity associated with the body is gone, and beyond that point the body won’t go any up further. This is because, as we all know, the kinetic energy is a function of velocity. At the maximum height, this energy is reflected as potential energy of the body, for which the body will now fall purely under the influence of gravity. Some part of energy is lost at reaching the highest point due to air resistance.
The potential energy of the body is given by the following expression:
\[PE = mgh\] …… (2)
Where,
\[m\] indicates mass of the body.
\[g\] indicates acceleration due to gravity.
\[h\] indicates maximum height.
Substituting the required values in the equation (2), we get:
\[ PE = mgh \\
 \Rightarrow PE = 1 \times 10 \times 18 \\
  \Rightarrow PE = 180\,{\text{J}} \\
\]
The potential energy is found out to be \[180\,{\text{J}}\]
Now the energy lost due to air resistance is the difference between kinetic energy and potential energy.
\[
  {\text{loss}}\,{\text{of}}\,{\text{energy}} = KE - PE \\
 \Rightarrow {\text{loss}}\,{\text{of}}\,{\text{energy}} = \left( {200 - 180} \right)\,{\text{J}} \\
 \Rightarrow {\text{loss}}\,{\text{of}}\,{\text{energy}} = 20\,{\text{J}} \\
\]
Hence, the energy lost due to air resistance is \[20\,{\text{J}}\] .

Note: This problem is based on the conversion from one form to another. Energy can neither be created nor be destroyed but can only be converted from one form to the other. In this case, some part of energy is also lost to overcome the friction provided by air. Always remember that final energy is less than that of initial energy as no external force continues to be exerted on the body during the upward motion.