At what speed should a space probe be fired from the earth if it is required to still be traveling at a speed of 3.48 km/s, even after coasting to an exceedingly great distance from the planet (a distance that is essentially infinite)?
Figure the work it takes to move an object from Earths surface to infintity, then add the remaining KE you want.
Work=INT GMe*m dr/r^2 + m 3.48^2
Set that equal to the KE at Earths Surface, mv^2, and solve for v.
To determine the required speed at which a space probe should be fired from Earth in order to maintain a velocity of 3.48 km/s after coasting to an infinite distance, we need to consider the concept of escape velocity.
The escape velocity is the minimum speed an object must achieve to escape the gravitational pull of a planet or celestial body. The escape velocity is given by the equation:
v = √(2 * g * R),
where v is the escape velocity, g is the acceleration due to gravity, and R is the radius of the planet.
For Earth, the gravitational acceleration is approximately 9.8 m/s^2, and the radius is approximately 6,371 km (or 6,371,000 meters).
Converting the final velocity of 3.48 km/s to meters per second gives us 3,480 m/s.
To find the velocity at launch, we should solve the equation for the escape velocity, substituting v with 3,480 m/s, g with 9.8 m/s^2, and R with 6,371,000 meters:
3,480 = √(2 * 9.8 * 6,371,000).
Squaring both sides of the equation:
12,110,400 = 2 * 9.8 * 6,371,000.
Dividing both sides by 19.6:
617,377.55 = 6,371,000.
Taking the square root of 617,377.55:
v = √617,377.55 ≈ 785.76 m/s.
Therefore, the space probe should be fired from Earth at a speed of approximately 785.76 m/s if it needs to maintain a velocity of 3.48 km/s after coasting to an infinite distance.