Consider the same asteroid initially in the asteroid belt at a distance 3 AU from the Sun, and 2 AU from Earth, which collides with Earth. As it falls towards Earth, it acquires kinetic energy (neglect its initial motion) both by being accelerated towards the Sun and towards the Earth. Find the change in its potential energy due to the Sun’s gravitational force, in megatons. Round to two significant digits.
To find the change in potential energy of the asteroid due to the Sun's gravitational force, we need to calculate the initial and final potential energies and then find their difference.
1. Find the initial potential energy (PEi) of the asteroid at a distance of 3 AU from the Sun using the formula:
PEi = -GMm / Ri,
where G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2), M is the mass of the Sun (1.989 × 10^30 kg), m is the mass of the asteroid, and Ri is the initial distance of the asteroid from the Sun in meters.
Converting 3 AU to meters:
Ri = 3 AU * 1.496 × 10^11 m/AU = 4.488 × 10^11 m.
Substituting the values into the formula:
PEi = -(6.67430 × 10^-11 N m^2/kg^2) * (1.989 × 10^30 kg) * m / (4.488 × 10^11 m).
2. Find the final potential energy (PEf) of the asteroid at a distance of 1 AU from the Sun (after falling towards Earth). Using the same formula:
Ri = 1 AU * 1.496 × 10^11 m/AU = 1.496 × 10^11 m.
PEf = -(6.67430 × 10^-11 N m^2/kg^2) * (1.989 × 10^30 kg) * m / (1.496 × 10^11 m).
3. Calculate the change in potential energy (ΔPE) by subtracting the initial potential energy from the final potential energy:
ΔPE = PEf - PEi.
Now, we need to determine the mass of the asteroid to calculate the potential energy change.
To find the change in potential energy due to the Sun's gravitational force, we need to calculate the initial and final potential energy of the asteroid.
Step 1: Calculate the initial potential energy
The initial potential energy of the asteroid at a distance of 3 AU from the Sun can be calculated using the gravitational potential energy formula:
PE_initial = -G * (Mass of the Sun) * (Mass of the asteroid) / (initial distance between the Sun and the asteroid)
Given that the distance between the Sun and the asteroid is 3 AU, we can convert it to meters:
Initial distance = 3 AU * (149,597,870,700 meters / 1 AU)
Initial distance = 448,793,612,100 meters
We'll also need the masses of the Sun and the asteroid. The mass of the Sun is approximately 1.989 x 10^30 kg, and the mass of the asteroid can be assumed to be negligible compared to the Sun's mass in this calculation.
Using the value of the gravitational constant, G = 6.67430 x 10^-11 m^3/kg/s^2, we can now calculate the initial potential energy.
PE_initial = - (6.67430 x 10^-11 m^3/kg/s^2) * (1.989 x 10^30 kg) * (Mass of the asteroid) / (448,793,612,100 meters)
Step 2: Calculate the final potential energy
The final potential energy of the asteroid when it collides with Earth can be calculated using the same formula, considering the final distance between the Sun and the asteroid as the distance between Earth and the Sun (1 AU = 149,597,870,700 meters).
Final distance = 149,597,870,700 meters
PE_final = - (6.67430 x 10^-11 m^3/kg/s^2) * (1.989 x 10^30 kg) * (Mass of the asteroid) / (149,597,870,700 meters)
Step 3: Calculate the change in potential energy
The change in potential energy is given by:
Change in PE = PE_final - PE_initial
Finally, we need to convert the Change in PE from joules to megatons. To do this, we divide the value by the energy released by the explosion of 1 megaton of TNT, which is approximately 4.184 x 10^15 joules.
Change in PE (megatons) = (Change in PE / 4.184 x 10^15 joules) * (1 megaton / 10^6 tons)
Round the calculated value to two significant digits to get the final result.
Remember to substitute the mass of the asteroid in kilograms to get an accurate result.