Final answer:
An object must travel at approximately 91.4% the speed of light for its total energy to be 70% more than its rest energy.
Step-by-step explanation:
The question asks at what velocity must an object travel for its total energy to be 70% more than its rest energy. Utilizing the principles of special relativity and relativistic mechanics, we can calculate this. When an object's kinetic energy is 150% of its rest mass energy, it indicates that the total energy is 250% of the rest mass energy (100% rest energy + 150% kinetic energy). To find a situation where total energy is 170% of rest energy, we will look for kinetic energy being 70% of the rest energy. This is a slightly different scenario and needs careful analysis using the relativistic energy equation:
Etotal = \( \gamma m c^2 \), where \( \gamma \) is the Lorentz factor \( \gamma = \frac{1}{\sqrt{1 - (v/c)^2}} \).
The Lorentz factor increases as v approaches c, but it does not actually reach c. From the question, we want to find when Etotal is 170% the rest mass energy Erest, i.e., when \( \gamma \) is 1.7. Solving the Lorentz factor equation for v/c when \( \gamma \) is 1.7, we find that v is approximately 0.914c, which answer (a) highlights. Therefore, an object must travel at about 91.4% the speed of light to have its total energy be 70% more than its rest energy.