6.4 Kinetic energy

WHAT IS ENERGY?

Energy is the capacity to perform work. Energy can neither be created nor destroyed, and it can only be transformed from one form to another. The unit of Energy is the same as of work, i.e. Joules and its \(\text { dimension is, }\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\). All forms of energy are either kinetic or potential. The energy in motion is known as Kinetic Energy, whereas Potential Energy is the energy stored in an object and is measured by the amount of work done.

KINETIC ENERGY

If an object of mass \(m\) has velocity \(\vec{v}\), its kinetic energy \(K\) is
\(
K=\frac{1}{2} m \vec{v} \cdot \vec{v}=\frac{1}{2} m v^{2}  \dots (6.2)
\)
Kinetic energy is a scalar quantity. The kinetic energy of an object is a measure of the work an object can do by the virtue of its motion.

The kinetic energy of a system of particles is the sum of the kinetic energies of all its constituent particles,

i.e., \(K=\sum_{i} \frac{1}{2} m_{i} v_{i}^{2}\)

Example 6.7:

In a ballistics demonstration a police officer fires a bullet of mass \(50.0 \mathrm{~g}\) with speed \(200 \mathrm{~m} \mathrm{~s}^{-1}\) on soft plywood of thickness \(2.00 \mathrm{~cm}\). The bullet emerges with only \(10 \%\) of its initial kinetic energy. What is the emergent speed of the bullet?

Solution: The initial kinetic energy of the bullet is \(m v^{2} / 2=1000 \mathrm{~J}\). It has a final kinetic energy of \(0.1 \times 1000=100 \mathrm{~J}\). If \(v_{f}\) is the emergent speed of the bullet,
\(
\begin{array}{l}
\frac{1}{2} m v_{f}^{2}=100 \mathrm{~J} \\
v_{f}=\sqrt{\frac{2 \times 100 \mathrm{~J}}{0.05 \mathrm{~kg}}} \\
=63.2 \mathrm{~m} \mathrm{~s}^{-1}
\end{array}
\)

The speed is reduced by approximately \(68 \%\) (not \(\mathbf{9 0} \%\) ).