Kinetic Energy

Kinetic energy is the energy possessed by objects in motion. It varies directly to the mass of the object and the square of its velocity. Mathematically, it is expressed as

\(KE=\frac12 mv^2\)

where

              \(KE=\text{kinetic energy, in Joules}\;(J)\\ m=\text{mass, in kilograms}\;(kg)\\ v=\text{velocity, in meters per second}\;(m/s)\).

Since an object's mass has only positive value and that the square of a number regardless of the sign comes up with a positive answer, thus, kinetic energy can only have two values - positive and zero. An object in motion will always have positive kinetic energy and an object at rest or not moving has zero kinetic energy.

The work done by a force which causes an object to move is equal to the change in the kinetic energy of the object. Thus, kinetic energy can also be expressed as

\(\Delta KE=W \quad\quad\quad\quad\quad\quad; W=F\cdot \Delta d\\ \)

                   \(\quad\quad\;\;=F\cdot \Delta d \quad\quad\quad\quad;F=ma\;\text{and}\;\Delta d=\frac {{{v_{f}}^2} - {v_i}^2}{2a}\\ \)

                                                                                 \(\quad\quad\;\;=(ma) \begin{pmatrix} \frac {{{v_{f}}^2} - {v_i}^2}{2a} \end{pmatrix} \)

                                                                                 \(\quad\quad\;\;=(m) \begin{pmatrix} \frac {{{v_{f}}^2} - {v_i}^2}{2} \end{pmatrix} \).


Example 1.

Calculate the kinetic energy of a 48-kg object moving at 35 m/s.

Solution:

\(KE=\frac 12 mv^2\)

\(KE=\frac 12(48\;kg)(35\;m/s)^2=2.94 \times 10^4\;J \).


Example 2.

A 714.4-kg race car begins the race at a speed of 70 m/s. In the middle of the race, it is running at 102.8 m/s. Solve for the change in kinetic energy of the car.

Given:

\(m=714.4\;kg\\ v_i=70\;m/s\\ v_f=102.8\;m/s\)

Solution:

To solve for the change in the kinetic energy, we will be using the formula \(KE=(m) \begin{pmatrix} \frac {{{v_{f}}^2} - {v_i}^2}{2} \end{pmatrix} \).

\(KE=(714.4\;kg) \begin{pmatrix} \frac {{102.8^2} - 70^2}{2}\end{pmatrix}\frac{m^2}{s^2}\\ \quad\;\;=(714.4\;kg) \begin{pmatrix}2,833.92\;\frac{m^2}{s^2}\end{pmatrix}\\ \quad\;\;=2.02\times 10^6\;J\).

 

Therefore, the change in kinetic energy of the race car is \(2.02\times 10^6\;J\).