Question

How to prove the equation of kinetic energy ($KE=\dfrac{1}{2}m{{v}^{2}}$)?

Answer 1

We need to understand the origin of the given equation for the kinetic energy in order to find the proper method of derivation of this given formula for the kinetic energy. This understanding will help us solve the given problem easily.

Complete answer:
We know that the kinetic energy is the form of mechanical energy possessed by a body by virtue of its motion. A body at rest has a kinetic energy equal to zero and the complete mechanical energy is stored as the potential energy of the object.
We can derive the kinetic energy very easily using Newton's second law of motion and the equations of motion. According to the Newton’s law of motion, the force applied on a body of mass ‘m’ can produce an acceleration which can be given as –

& F=ma \\\ & \therefore a=\dfrac{F}{m} \\\ \end{aligned}$$ This acceleration exists as long as the force is present on the body. We know that the three equations of motion are – $$\begin{aligned} & v=u+at \\\ & S=ut+\dfrac{1}{2}a{{t}^{2}} \\\ & {{v}^{2}}-{{u}^{2}}=2as \\\ \end{aligned}$$ Where, v is the final velocity of the moving object, u is the initial velocity of the moving object, a is the acceleration of the object, t is the time taken for motion and S is the distance travelled by the body. Let us consider the third equation of motion as it is time-dependent to prove the kinetic energy equation. Consider a body initially at rest applied with a force producing an acceleration. Also, we know that the kinetic energy and the force are related by the displacement as – $$\begin{aligned} & KE=FS \\\ & \Rightarrow S=\dfrac{KE}{F} \\\ & \therefore S=\dfrac{KE}{ma} \\\ \end{aligned}$$ So, we can substitute the known quantities in the third equation of motion as – $$\begin{aligned} & {{v}^{2}}-{{u}^{2}}=2as \\\ & \Rightarrow {{v}^{2}}-{{u}^{2}}=2a\dfrac{KE}{ma} \\\ & \text{also, }u=0 \\\ & \Rightarrow 2KE=m{{v}^{2}} \\\ & \therefore KE=\dfrac{1}{2}m{{v}^{2}} \\\ \end{aligned}$$ Thus, we proved the equation of kinetic energy. This is the required solution. **Note:** We can prove the kinetic energy equation through many methods. We can consider a force which opposes the motion and turning the final velocity to zero, which will give the initial kinetic energy equal to what we have just found or we can use the integration method also.