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Welcome to our Physics lesson on Impulse in Relativistic Events, this is the second lesson of our suite of physics lessons covering the topic of Relativistic Dynamics. Mass, Impulse and Energy in Relativity, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Despite that the Newton's Second Law written as
is OK, we cannot rely anymore on the Newtonian relation p⃗ = m ∙ v⃗ which assumes the mass as constant. For this reason, we must find a new form of the impulse expression. This is found by considering the process of elastic collision between two small spheres. From tutorial 6.6 "Collision and Impulse. Types of Collision", we know that in (perfectly) elastic collisions, the kinetic energy and impulse are both conserved, i.e. the impulse (and kinetic energy) of the system before and after the collision is equal. This concept is valid in the relativistic events as well. If we analyze the law of conservation of impulse, we find that a constant Newtonian mass does not necessarily result in a conservation of impulse.
Since the length of objects in relativistic events experiences contraction, there must be an increase in mass at the same rate (recall that the shape of objects does not change - a sphere will still be a sphere, a cube will still be a cube after relativistic contraction and so on). The rate of mass increase is the equal to that of length decrease. Therefore, we obtain the following equation for the relativistic mass
where m is the relativistic mass of particle, m0 is the classical mass (rest mass), and v is the velocity of particle in the system S. This means we obtain for the relativistic impulse
This formula still contains the relationship between impulse, mass and velocity but now this relationship is not anymore linear. The above two formulae are closely related to each other and are known as the relativistic formulae of mass and impulse. It is worth to mention here that these formulae are valid for all objects whether microscopic or macroscopic. (In the practical sense, many microscopic particles move very often at very high speeds, comparable to the speed of light). The concept of Newtonian mass m0 is very easy to understand. If v = 0, then m0 = m. This means the Newtonian mass m0 (otherwise known as the rest mass m0) is nothing else but the relativistic mass of particle in an inertial system where the particle is at rest. For v → c the mass and impulse point towards infinity (m → ∞, p → ∞). Hence, the velocity v = c is as a barrier for material particles.
A proton has a rest mass m0 and a velocity of v = 0.5c. How much percent does the mass increase during this motion when compared to the rest mass?
We must find the relativistic mass m given the rest mass m0 and velocity v.
Using the equation of relativistic mass
we obtain for the mass m of particle after substituting the values:
Thus, the new mass is
of the original mass. This results in an increase of
in the mass of particle.
You have reached the end of Physics lesson 18.6.2 Impulse in Relativistic Events. There are 3 lessons in this physics tutorial covering Relativistic Dynamics. Mass, Impulse and Energy in Relativity, you can access all the lessons from this tutorial below.
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