The fastest vehicle flown by humans on a regular basis is...

The fastest vehicle flown by humans on a regular basis is the space shuttle which has a mass of about $1 \times 10^{5} \mathrm{~kg}$ and travels at a speed of about $8 \times 10^{3} \mathrm{~m} / \mathrm{s}$. Find the momentum of the space shuttle using relativistic formulae and then again by using nonrelativistic formulae. By what percent do the relativistic and nonrelativistic momenta of the space shuttle differ? [Hint: You might want to use one of the approximations in Appendix A.5.]

Key Concepts

Non-Relativistic Momentum

Non-relativistic momentum is determined by the simple product p = mv, which assumes that the object's speed is much lower than the speed of light. This approximation neglects relativistic effects and is accurate only when the velocities involved are low compared to c.

Lorentz Factor Approximation

The Lorentz factor, ? = 1/?(1 - v²/c²), can be approximated for small v/c using a Taylor expansion, resulting in ? ? 1 + ½(v/c)². This approximation highlights that when v is much smaller than c, the relativistic momentum converges closely to the non-relativistic momentum, making relativistic corrections minimal.

Relativistic Momentum

Relativistic momentum is calculated using the formula p = ?mv, where ? is the Lorentz factor defined as 1/?(1 - v²/c²). This formulation accounts for the effects of special relativity, ensuring accurate momentum values even when the speed of the object is a significant fraction of the speed of light.

Percentage Difference

The percentage difference is a measure of the relative error between two calculated values. It is obtained by finding the absolute difference between the relativistic and non-relativistic momentum values, dividing by the relativistic momentum (or a reference value), and then multiplying by 100. This quantifies how much the non-relativistic approximation deviates from the exact relativistic result.

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