Help Me With Astronomy Assignment And Quantitative Reasoning

Help Me With Astronomy Assignment Please Bquantitative Reason

Help me with Astronomy assignment please!! B. Quantitative reasoning a. How much energy is produced when the sun converts 1 kg of hydrogen into helium? If the average household uses 1400 kW-hr per month, how long would the conversion of 1 kg of hydrogen to helium power an average household? b. If you are cruising down the road at 50 mph and a policeman with a radar gun (wavelength 1 cm) reflects radar light off your bumper, what wavelength of radar light does he get back? Assume the policeman is stationary, and you are moving towards him. Light that gets reflected has twice as much shift as light that is just emitted, so use the formula for the shift of emitted light but multiply by two. (Hint: convert mph to m/s.)

Paper For Above instruction

Understanding the principles of energy production in stars and the physics of electromagnetic waves is critical in astrophysics and related disciplines. In this paper, we will analyze the energy released during nuclear fusion in the Sun, specifically the fusion of hydrogen into helium, and explore the application of the Doppler effect in radar technology used for speed detection. The integration of these topics highlights the importance of quantitative reasoning in comprehending astrophysical phenomena and real-world applications involving wave physics.

Energy Produced by Fusion of Hydrogen into Helium in the Sun

Nuclear fusion in the Sun primarily involves the conversion of hydrogen nuclei into helium, releasing a significant amount of energy in the process. The dominant reaction can be summarized as:

4 ^1H → ^4He + 2 e⁺ + 2 ν_e + energy

Calculating the energy produced involves understanding the mass difference and Einstein's mass-energy equivalence (E=mc²). The mass of four hydrogen nuclei is approximately 4.032 u, while a helium nucleus has a mass of about 4.0015 u, resulting in a mass defect of about 0.0305 u. Using the conversion 1 u = 931.5 MeV/c², the energy released per fusion event is:

ΔE = 0.0305 u × 931.5 MeV/u ≈ 28.4 MeV

Converting MeV to Joules (1 eV = 1.602 × 10⁻¹⁹ Joules, so 1 MeV = 1.602 × 10⁻¹³ Joules):

ΔE ≈ 28.4 × 1.602 × 10⁻¹³ J ≈ 4.55 × 10⁻¹² J

This is the energy released from the fusion of 4 hydrogen nuclei, equivalent to approximately 2.26 × 10⁻¹² Joules per hydrogen atom. Since 1 kg of hydrogen contains approximately 6.02 × 10²⁶ atoms (using Avogadro's number and molar mass), the total energy produced by converting 1 kg of hydrogen is:

Total energy = (number of atoms) × (energy per atom)

Total energy ≈ 6.02 × 10²⁶ × 2.26 × 10⁻¹² J ≈ 1.36 × 10¹⁵ Joules

How Long Can This Power a Household?

The average household consumes about 1400 kWh per month. To compare, convert kilowatt-hours to Joules:

1 kWh = 3.6 × 10⁶ Joules

Therefore, monthly consumption in Joules is:

1400 kWh × 3.6 × 10⁶ J/kWh = 5.04 × 10⁹ Joules

Using the total energy from 1 kg of hydrogen (~1.36 × 10¹⁵ Joules), the duration this energy could power the household is:

Duration = Total energy / Monthly consumption

≈ 1.36 × 10¹⁵ J / 5.04 × 10⁹ J ≈ 2.7 × 10⁵ months

Converting months to years:

≈ 2.7 × 10⁵ months / 12 ≈ 22500 years

Hence, the fusion of 1 kg of hydrogen could theoretically power an average household for approximately 22,500 years.

Doppler Shift in Radar Reflection from a Moving Car

The Doppler effect causes a shift in the observed wavelength of waves reflected from moving objects. The fundamental formula for the Doppler shift when the source or reflector is moving towards the observer (stationary radar gun) is:

λ_observed = λ_emitted × (1 - v/c)

where v is the velocity of the moving object, and c is the speed of light (~3 × 10⁸ m/s). First, convert the speed from miles per hour to meters per second:

50 mph ≈ 50 × 0.44704 m/s ≈ 22.35 m/s

Since the radar gun reflects the wave, and reflection doubles the Doppler shift, the effective shift factor becomes:

Factor = 2 × (v/c)

Calculating this:

2 × (22.35 / 3 × 10⁸) ≈ 1.49 × 10⁻⁷

Next, find the wavelength of the emitted radar wave, which is given as 1 cm = 0.01 m. The reflected wavelength λ_reflected is then:

λ_reflected = λ_emitted × (1 - 2 × v/c)

Plugging in the numbers:

λ_reflected = 0.01 m × (1 - 1.49 × 10⁻⁷) ≈ 0.01 m × 0.99999985 ≈ 0.0099999985 m

The change is minimal, but the reflected wave experiences a slight reduction in wavelength due to the Doppler effect, resulting in a wavelength of approximately 9.9999985 mm.

In practice, this tiny change is detectable with sensitive radar equipment, allowing precise speed measurements.

Conclusion

The analysis demonstrates the immense energy released during stellar nucleosynthesis, enough to sustain a household for tens of thousands of years per kilogram of hydrogen. Moreover, understanding the Doppler effect's application in radar technology underscores its importance in everyday life, such as traffic speed enforcement. These concepts reinforce the significance of quantitative reasoning to interpret physical phenomena accurately and practically.

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