Final answer:
To find the speed at which relativistic and Newtonian kinetic energies differ by 10%, one would compare the classical kinetic energy expression (KEclass = 1/2 mv^2) with the relativistic formula (KErel = (γ - 1)mc^2) where γ is the Lorentz factor. The speed can be found by setting KErel to be 10% greater than KEclass and solving for v/c, which requires the application of algebra.
Step-by-step explanation:
To answer the question of at what speed the relativistic and Newtonian expressions for kinetic energy differ by 10%, we have to compare the two expressions for kinetic energy. The Newtonian or classical kinetic energy (KEclass) is given by the expression KEclass = 1/2 mv2, while the relativistic kinetic energy (KErel) involves the Lorentz factor (γ) and is given by KErel = (γ - 1)mc2, where γ = 1 / √(1 - (v/c)2).
First, we need to establish a condition for when KErel is 10% greater than KEclass:
KErel = 1.10 * KEclass. From this, we can derive a relationship involving v, the velocity of the particle, and c, the speed of light. Solving this equation for v/c will give us the velocity as a fraction of the speed of light.
To arrive at the precise speed where this 10% difference occurs, detailed calculations involving algebra are necessary, which would fall under a more advanced physics problem typically dealt with in higher levels of education, such as college physics courses.
Nevertheless, it is important to note that as the velocity approaches a significant fraction of the speed of light, the relativistic effects become non-negligible, and the classical kinetic energy underestimates the actual kinetic energy of the object.