Final answer:
The momentum of a 1.00×109 kg asteroid traveling at 30.0 km/s is 3.00×1013 kg·m/s. The ratio of this momentum to the classical momentum considering relativistic effects is 1.000000005, indicating negligible relativistic effects due to the asteroid's velocity being much lower than the speed of light.
Step-by-step explanation:
To find the momentum of a 1.00×109 kg asteroid heading towards the Earth at 30.0 km/s, we use the classical formula: p = m × v. Therefore, the momentum (p) is:
p = (1.00×109 kg) × (30.0 km/s)
p = (1.00×109 kg) × (30,000 m/s) — converting km/s to m/s
p = 3.00×1013 kg·m/s
To find the ratio of this momentum to the classical momentum when considering relativistic effects, we first use the approximation: γ = 1 + (1/2)v²/c², where c is the speed of light. Since the asteroid's velocity is quite small compared to the speed of light, the value of γ is very close to 1. The ratio is thus:
pratio = γ
pratio = 1 + (1/2)(30,000 m/s)²/(3×108 m/s)² — using the value for c as the speed of light in vacuum
pratio = 1 + 0.000000005
pratio = 1.000000005 (to show the small relativistic effects)
The correct answer is (a) 3.00×1013 kg·m/s for the momentum and (b) 1.000000005 for the pratio.