Final answer:
George must increase his speed by a factor of the square root of 2 to have the same kinetic energy as Oliver because kinetic energy is proportional to the square of the velocity and directly proportional to the mass.
Step-by-step explanation:
The question is about how much George needs to speed up to have the same kinetic energy as Oliver, given that Oliver is twice as fast but half as massive as George. Kinetic energy (KE) is given by the formula KE = 0.5 × m × v², where m is mass and v is velocity. Since kinetic energy is proportional to the square of the velocity, we can analyze the relationship between mass and velocity to understand the kinetic energies of both Oliver and George.
Let's denote George's mass as M and his speed as V. According to the question, Oliver's mass would be M/2 and his speed 2V. Since Oliver's speed is squared in the kinetic energy formula, doubling his speed results in four times the kinetic energy related to the speed component. However, because Oliver's mass is half that of George, his kinetic energy becomes twice that of George if George keeps the same speed, because KE is directly proportional to mass.
To have the same kinetic energy as Oliver, George must overcome the effect of the doubled speed resulting in quadrupled energy due to the velocity's contribution to kinetic energy. Therefore, George needs to increase his speed by a factor of √2 (√4 = 2) to make up for both the difference in mass and the initially higher speed of Oliver. Therefore, to have the same kinetic energy as Oliver, George must speed up by a factor of √2 of his original speed.