In the Einstein’s theory of special relativity, the relativistic mechanics is conce rned with the motion of bodies whose vel ocities approach the speed of light . This paper presents the Einstein’s mass -energy equivalence and the equations of relativistic mass , momentum and energy from the Newton’s second law of motion. Keywords: relativistic motion , momentum equation, kinet ic energy, rest mass 1. Introduction The equation of Einstein’s mass -energy equivalence [1-3] is 𝐸=𝑚𝑐2, where E, m, and c denote electromagnetic/light energy, mass of light, and speed of light respectively. The derivation of the equation s of relativistic mass and mass -energy equivalence is obtained from the Newton’s second law of motion by diffe rentiation and integration [4]. 2. Relativistic Mass, Momentum and Energy In the Einstein’s theory of speci al relativity, the relativistic momentum is concerned with the motion of particle whose velocity approaches the speed of light . Let us derive the re lativistic mass , momentum , and energy as follows: The Newton’s second law of motion states that the force ( F) acting on a particle is equal to the rate of change of its momentum ( p). 𝐹=𝑑𝑝 𝑑𝑡=𝑑(𝑚𝑣) 𝑑𝑡=𝑚𝑑𝑣 𝑑𝑡+𝑣𝑑𝑣 𝑑𝑡,where 𝑣 is velocity. The differential equation for work and kinetic energy is derived as follows : 𝑑𝐾=𝑑𝑊=𝐹𝑑𝑠 𝑑𝐾=𝐹𝑑𝑠=(𝑚𝑑𝑣 𝑑𝑡+𝑣𝑑𝑣 𝑑𝑡 )𝑑𝑠 𝑑𝐾=𝐹𝑑𝑠=𝑚𝑑𝑠 𝑑𝑡𝑑𝑣+𝑣𝑑𝑠 𝑑𝑡𝑑𝑣,where 𝑑𝑠 𝑑𝑡=𝑣 The kinetic energy per second is denoted by 𝑑𝐾, which is equivalent to 𝑐2𝑑𝑚. The variable mass of a moving particle at a second is denoted by 𝑑𝑚. 𝑑𝐾=𝑚𝑣𝑑𝑣+𝑣2𝑑𝑚 𝑐2𝑑𝑚=𝑚𝑣𝑑𝑣+𝑣2𝑑𝑚 Page | 2 𝑑𝑚 𝑚=𝑣 𝑐2−𝑣2𝑑𝑣 ∫𝑑𝑚 𝑚𝑚 𝑚0=∫𝑣 𝑐2−𝑣2𝑑𝑣𝑣 0 [ln(𝑚)]𝑚0𝑚=−1 2[ln(𝑐2−𝑣2)]0𝑣 ln𝑚−ln𝑚0=−1 2ln(𝑐2−𝑣2)+1 2 ln𝑐2 ln𝑚 𝑚0=1 2ln𝑐2 𝑐2−𝑣2 𝑚 𝑚0=√𝑐2 𝑐2−𝑣2 Relativistic mass (𝑚)=𝑚0 √1−𝑣2 𝑐2 Where,  the rest mass of the body is 𝑚0  the velocity of the body in motion is 𝑣  the speed of the light is 𝑐 Relativistic momentum (𝑝)=𝑚0𝑣 √1−𝑣2 𝑐2 Relativistic Energy (𝐸)=𝑚0𝑐2 √1−𝑣2 𝑐2 The relation between relativisti c energy and momentum show below : 𝐸2=𝑚02𝑐4 1−𝑣2 𝑐2⟹𝐸2=𝑚02𝑐2(𝑣2−𝑣2+𝑐2) 1−𝑣2 𝑐2⟹𝐸2=𝑚02𝑐2𝑣2−𝑚02𝑐2𝑣2+𝑚02𝑐4 1−𝑣2 𝑐2 𝑇ℎ𝑒𝑛,𝐸2= ( 𝑚0𝑣 √1−𝑣2 𝑐2) 2 𝑐2+𝑚02𝑐2(𝑐2−𝑣2) 𝑐2−𝑣2 𝑐2 From the above expression, we obtain the energy -momentum relation 𝐸2=𝑝2𝑐2+𝑚02𝑐2. If particle is at rest, then 𝑝=0. Thus, the rest energy is that 𝐸=𝑚0𝑐2. Page | 3 Now, it is understood that the relativistic mass, momentum, and energy are derived from the Newton’s second law of motion. 3. Mass -Energy Equiv alence Relativistic mass (𝑚)=𝑚0 √1−𝑣2 𝑐2 𝑚2=𝑚02 1−𝑣2 𝑐2⟹𝑚2(𝑐2−𝑣2)=𝑚02𝑐2,where rest mass energy=𝑚02𝑐2 𝑚2𝑐2−𝑚2𝑣2=𝑚02𝑐2 By differentiating the equation, we get 2𝑚𝑐2𝑑𝑚−2𝑚𝑣2𝑑𝑚−𝑚22𝑣𝑑𝑣=0, where the rest mass ( 𝑚0) and the speed of light ( 𝑐) are constant . The kinetic energy ( 𝑑𝐾) is equivalent to 𝑐2𝑑𝑚, where c is the speed of light. 𝑑𝐾=𝑐2𝑑𝑚 ∫𝑑𝐾𝐾 0=∫𝑐2𝑑𝑚𝑚 𝑚0 𝐾=𝑐2(𝑚−𝑚0),where 𝐾 is kinetic energy. Total Energy (𝐸) = Kinetic Energy (𝐾) + Rest Mass -Energy (𝑚0𝑐2) 𝐸=𝑐2(𝑚−𝑚0)+𝑚0𝑐2 𝐸=𝑐2𝑚−𝑐2𝑚0+𝑚0𝑐2 𝐸=𝑐2𝑚 Therefore ,𝐸=𝑚𝑐2. Hence, the equa tion of Einstein’s mass -energy equivalence is derived from Newton’s Second Law of Motion . 4. Conclusion The mass -energy equivalence along with relativistic mass , momentum , and energy play a n important role in the Einstein ’s theory of special relativity . Also, the relativistic mechanics is conce rned with the motion of bodies whose vel ocities approach the speed of light . In this article , the equation of Einstein’s mass -energ y equivalence and the relativistic mass and momentum equation have been derived from Newton’s Second Law of Motion .