The aim of this article is to find the Kinetic Energy of an object in motion in $BTU$.
The basic concept behind this article is the understanding of Kinetic Energy K.E. and its unit conversion.
Kinetic Energy is defined as the energy that an object carries while in motion. All moving objects possess kinetic energy. When a net force $F$ is applied to an object, this force transfers energy, and resultantly work $W$ is done. This energy called Kinetic Energy K.E. changes the state of the object and causes it to move at a certain speed. This Kinetic Energy K.E. is calculated as follows:
\[Work\ Done\ W\ =\ F\ \times\ d\]
Where:
$F\ =$ Net Force Applied to the Object
$d\ =$ Distance traveled by the Object
Since:
\[F\ =\ m\ \times\ a\]
So:
\[W\ =\ (m\ \times\ a)\ \times\ d\]
As per the Equation of Motion:
\[2\ a\ d\ =\ {v_f}^2\ -\ {v_i}^2\]
And:
\[a\ =\ \frac{{v_f}^2\ -\ {v_i}^2}{2d}\]
Substituting in the equation for work done, we get:
\[W\ =\ m\ \times\ d\ \times\ \left(\frac{{v_f}^2\ -\ {v_i}^2}{2d}\right)\]
\[W=\frac{1}{2}\ m\times({v_f}^2\ -\ {v_i}^2)\]
If the object is initially at rest, then $v_i=0$. So, simplifying the equation, we get:
\[K.E.\ \ =\ \frac{1}{2}\ m\ {\ v}^2\]
Where:
$m$ is the mass of the object, and $v$ is the velocity of the object.
The SI Unit for Kinetic Energy K.E. is Joules $J$ or $BTU$ (British Thermal Unit).
Expert Answer
Given that:
Mass of the Object $m\ =\ 10\ lbm$
Velocity of the Object $v\ =\ 50\ \dfrac{ft}{s}$
We need to find the Kinetic Energy K.E. which is calculated as follows:
\[K.E.\ \ =\ \frac{1}{2}\ m{\ v}^2\]
Substituting the given values in the above equation, we get:
\[K.E.\ \ =\ \frac{1}{2}\ (10\ lbm){\ (50\ \frac{ft}{s})}^2\]
\[K.E.\ \ =\ 12500\ lbm \frac{{\rm ft}^2}{s^2}\]
We need to calculate the Kinetic Energy K.E. in $BTU$ – British Thermal Unit.
As we know:
\[1\ BTU\ =\ 25037\ lbm \frac{{\rm ft}^2}{s^2}\]
\[1\ lbm \frac{{\rm ft}^2}{s^2}\ =\ \frac{1}{25037}\ BTU\]
Hence:
\[K.E.\ \ =\ 12500\ \times\ \frac{1}{25037}\ BTU\]
\[K.E.\ \ =\ 0.499\ BTU\]
Numerical Result
The Kinetic Energy of the Object in BTU is as follows:
\[K.E.\ \ =\ 0.499\ BTU\]
Example
If an object having a mass of $200kg$ is moving at the speed of $15\dfrac{m}{s}$, calculate its Kinetic Energy in Joules.
Solution
Given that:
Mass of the Object $ m\ =\ 200\ kg $
Velocity of the Object $ v\ =\ 15\ \dfrac{m}{s} $
We need to find the Kinetic Energy K.E. which is calculated as follows:
\[ K.E.\ \ =\ \frac{1}{2}\ m{\ v}^2 \]
Substituting the given values in the above equation, we get:
\[ K.E.\ \ =\ \frac{1}{2}\ (200\ kg){\ (15\ \frac{m}{s})}^2 \]
\[ K.E.\ \ =\ 22500\ kg\ \frac{m^2}{s^2} \]
As we know:
The SI unit of Kinetic Energy is Joule $J$ which is expressed as follows:
\[ 1\ Joule\ J\ =\ 1\ kg\ \frac{m^2}{s^2} \]
Hence:
\[ K.E.\ \ =\ 22500\ J \]
\[ K.E.\ \ =\ 22500\ \frac{J}{1000} \]
\[ K.E.\ \ =\ 22.5\ KJ \]