Suppose that the speed of light in a vacuum is only 45.0 meters per second. Calculate the relativistic kinetic energy of a car of mass 1.00 times 10 to the third kilograms that has a velocity of 30.0 meters per second. Find the ratio of the car’s relativistic kinetic energy to its classical kinetic energy.
We can call the new speed of light in vacuum 45.0 meters per second 𝑐, the mass of the car 1.00 times 10 to the third kilograms 𝑚, and the car’s speed 30.0 meters per second 𝑣. We want to solve first for the relativistic kinetic energy of the car. We can call that KE sub 𝑟. And second we wanna solve for the ratio of the relativistic kinetic energy of the car to its classical kinetic energy. We can call that KE sub 𝑟 to KE sub 𝑐.
Our car we’re told has a mass 𝑚 and a speed 𝑣, both given in the problem statement. To solve for its relativistic kinetic energy, we can recall the mathematical expression for that term. KE sub 𝑟 equals the quantity 𝛾 minus one times mass times the speed of light squared, where 𝛾 equals one divided by the square root of one minus 𝑣 squared over 𝑐 squared.
Applying this relationship to our situation, we’ve been given the car’s speed 𝑣, the speed of light 𝑐, and the car’s mass 𝑚. So we’re ready to plug in and solve for KE sub 𝑟. When we plug in each one of these terms and enter this expression on our calculator, we find that KE sub 𝑟 equals 6.92 times 10 to the fifth joules. That’s the car’s relativistic kinetic energy.
Now in order to find the ratio of relativistic to classical kinetic energy, let’s recall the mathematical relationship for classical KE. It’s equal to one-half an object’s mass times its speed squared. If we divide this car’s relativistic kinetic energy by its classical kinetic energy that equals 6.92 times 10 to the fifth joules divided by one-half 𝑚𝑣 squared, or plugging in for 𝑚 and 𝑣 and entering this fraction on our calculator, we find that, to three significant figures, it equals 1.54. So the ratio of relativistic to classical kinetic energy of this car is 1.54 to one.