If a projectile is launched from Earth with a speed equal to the escape speed, how high above the Earth's surface is it when its speed is one third the escape speed?

To find the height above the Earth's surface when the speed of the projectile is one third of the escape speed, we need to consider the concept of energy conservation and the formula for escape speed.

Let's begin by understanding escape speed. Escape speed is the minimum speed required for a projectile to overcome the gravitational pull of the Earth and escape its gravitational field. The formula for escape speed is given by:

v_escape = sqrt(2 * G * M / R)

Where:
- v_escape is the escape speed
- G is the gravitational constant (approximately 6.67430 × 10^(-11) m^3 kg^(-1) s^(-2))
- M is the mass of the Earth (approximately 5.972 × 10^24 kg)
- R is the radius of the Earth (approximately 6,371 km or 6,371,000 meters)

Now, we know that the projectile was launched with a speed equal to the escape speed. Let's call this initial speed v_initial.

So, v_initial = v_escape

Now, we're given that the speed of the projectile when it is at some height h above the Earth's surface is one third of the escape speed. Let's call this speed v_final.

So, v_final = (1/3) * v_escape

To find the height h, we need to use the principle of energy conservation which states that the total mechanical energy of the projectile is conserved throughout its motion.

The total mechanical energy of the projectile at any point is given by the sum of its kinetic energy (0.5 * m * v^2) and its gravitational potential energy (m * g * h), where m is the mass of the projectile and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Since energy is conserved, we can set the initial mechanical energy equal to the final mechanical energy:

(0.5 * m * v_initial^2) - (m * g * h_initial) = (0.5 * m * v_final^2) - (m * g * h_final)

Since we're assuming the projectile is on Earth's surface initially, h_initial = 0. Therefore, the equation becomes:

(0.5 * m * v_initial^2) - (m * g * 0) = (0.5 * m * v_final^2) - (m * g * h_final)

Now, we can simplify the equation and solve for h_final:

(0.5 * v_initial^2) = (0.5 * v_final^2) - (g * h_final)

Rearranging the equation:

h_final = ((0.5 * v_initial^2) - (0.5 * v_final^2)) / g

Now we can substitute the values for v_initial, v_final, and g into the equation to find the height h_final.