Final answer:
The correct answer to the question is option (d): (p = m × γ × v), (p_{ratio} = 1). The classical formula for momentum is sufficient for a satellite moving at 4.00 km/s, as relativistic effects are negligible, resulting in the ratio of relativistic to classical momentum being 1.
Step-by-step explanation:
The question concerns the physics principle of momentum, specifically relating to a satellite orbiting Earth. Momentum is classically defined as the product of an object's mass and its velocity, given by the formula p = m × v. However, at high speeds approaching the speed of light, relativistic effects become significant, which requires a correction to the classical momentum formula. The relativistic momentum incorporates the Lorentz factor (γ), resulting in the formula p = m × γ × v. The question also addresses the approximation of the Lorentz factor at low velocities as γ = 1 + ½(v²/c²), where c is the speed of light (approximately 3 × 10⁸ m/s).
In part (a), to find the momentum of a 2000 kg satellite orbiting at 4.00 km/s, we use the classical formula since the velocity is much less than the speed of light: p = m × v. Thus, the momentum p is 2000 kg × 4000 m/s = 8 × 10⁶ kg·m/s.
In part (b), to find the ratio of this momentum to the classical momentum, we would calculate γ using the provided approximation and multiply the classical momentum by this factor. However, at the satellite's velocity, the factor γ is so close to 1 that the ratio is effectively 1, meaning the relativistic momentum is virtually identical to the classical momentum.
Therefore, option (c) is incorrect because the ratio should be 1, not γ, and option (a) is incorrect because there's no need to multiply by γ or have γ as part of the answer for the ratio since it's approximately 1. Option (b) also misrepresents the ratio because it suggests the ratio is γ itself. The correct option is (d) because the momentum formula includes the Lorentz factor, but since γ is approximately 1, the ratio of relativistic momentum to classical momentum is 1 (p_{ratio} = 1).