Video Transcript
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 carโs relativistic kinetic energy to its classical kinetic energy.
We can call the new speed of light in vacuum 45.0 meters per second ๐, the mass of the car 1.00 times 10 to the third kilograms ๐, and the carโs speed 30.0 meters per second ๐ฃ. We want to solve first for the relativistic kinetic energy of the car. We can call that KE sub ๐. 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 ๐ to KE sub ๐.
Our car weโre told has a mass ๐ and a speed ๐ฃ, both given in the problem statement. To solve for its relativistic kinetic energy, we can recall the mathematical expression for that term. KE sub ๐ equals the quantity ๐พ minus one times mass times the speed of light squared, where ๐พ equals one divided by the square root of one minus ๐ฃ squared over ๐ squared.
Applying this relationship to our situation, weโve been given the carโs speed ๐ฃ, the speed of light ๐, and the carโs mass ๐. So weโre ready to plug in and solve for KE sub ๐. When we plug in each one of these terms and enter this expression on our calculator, we find that KE sub ๐ equals 6.92 times 10 to the fifth joules. Thatโs the carโs relativistic kinetic energy.
Now in order to find the ratio of relativistic to classical kinetic energy, letโs recall the mathematical relationship for classical KE. Itโs equal to one-half an objectโs mass times its speed squared. If we divide this carโs relativistic kinetic energy by its classical kinetic energy that equals 6.92 times 10 to the fifth joules divided by one-half ๐๐ฃ squared, or plugging in for ๐ and ๐ฃ 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.
