Here’s the link to my Universal Gravitational Potential Energy presentation, along with an accompanying note sheet!
If the Earth suddenly stopped revolving around the Sun, it would begin to fall directly toward the Sun due to the force of gravity. To estimate the speed at which the Earth would crash into the Sun, we can apply the principle of conservation of mechanical energy. The Earth’s initial mechanical energy consists solely of gravitational potential energy, which is converted into kinetic energy as it falls toward the Sun.
The formula for gravitational potential energy is:
U = -G * (M1 * M2) / r
where U is the potential energy, G is the gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2), M1 and M2 are the masses of the two objects (in this case, the Earth and the Sun), and r is the distance between their centers.
The mass of the Sun (M1) is approximately 1.989 x 10^30 kg, and the mass of the Earth (M2) is approximately 5.972 x 10^24 kg. The average distance between the Earth and the Sun is about 1.496 x 10^11 meters.
Plugging these values into the formula, we get:
U_initial = -G * (M1 * M2) / r U_initial ≈ -6.67430 x 10^-11 * (1.989 x 10^30 * 5.972 x 10^24) / (1.496 x 10^11) U_initial ≈ -5.310 x 10^33 J (joules)
As the Earth falls toward the Sun, its gravitational potential energy is converted into kinetic energy. When the Earth crashes into the Sun, its potential energy will be zero, and its mechanical energy will consist entirely of kinetic energy:
K_final = -U_initial
The formula for kinetic energy is:
K = 0.5 * m * v^2
K is kinetic energy, m is the object’s mass, and v is velocity. We can solve for the Earth’s velocity when it crashes into the Sun:
0.5 * M2 * v^2 = -U_initial v^2 = -2 * U_initial / M2 v = sqrt(-2 * U_initial / M2)
Plugging in the values:
v = sqrt(-2 * (-5.310 x 10^33 J) / (5.972 x 10^24 kg)) v ≈ 3.29 x 10^5 m/s
Thus, the Earth’s speed as it crashed into the Sun would be approximately 329,000 meters per second or 329 km/s. This is a rough estimate and assumes no other forces act on the Earth during its fall, which is not entirely realistic.