Orbit and Escape Velocity

• Throw a ball straight up in the air and it falls back down.

• Have your strongest friend throw the ball and it might take a fraction longer to return to Earth.

• Orbit is a path described by one body in its revolution around another.

• Orbit does not depend on vertical velocity, but on horizontal velocity.

Orbital Basics

• Fire a cannon, and the ball travels some distance sideways before hitting the ground.

• Fire the ball faster, and it will travel further before hitting the ground.

• If we put a stupidly powerful cannon on top of a really tall mountain, the ball might have so much horizontal velocity that the rate it fell towards the Earth’s surface equaled the rate at which the Earth’s surface curved away beneath it.

• The ball would be in orbit around the Earth.

Orbital basics

Isaac Newton’s idea

• International Space Station (ISS) orbits at ~350km above the surface.

• If the Earth were the size of a basketball, the ISS would be 12.8mm above the surface.

• Because it’s not particularly high, the ISS has to have a huge horizontal velocity, and orbits Earth every 90 minutes.

• a path described by one body in its revolution about another

Outer Space

• Escape velocity: The minimum velocity at which a receding object has its kinetic energy equal to or greater than its gravitational potential energy due to your height above the Earth.

• Can be calculated by:

• (negative because it’s in the opposite direction).

• The m’s cancel, so • Asking a 6th grader, we find:

Escape Velocity

• On the Earth’s surface, we are about 6,400km from the center of Earth’s mass.

• So h=6,400,000m• g =9.81m/s2

• 2*9.81*6,400,000=125,568,000. Now just take the square root of that big number!

• v=11,205m/s, or 11.205km/s.

Escape Velocity - Earth

• To escape Earth’s gravity and not just fall in an endless circle, you need to go even faster: either 11.2km/s from the surface, or 10.9km/s from LEO.

• The drag of the atmosphere would make it very dangerous to accelerate a rocket to 11.2km/s at low altitude, so NASA puts probes into LEO first and then gives them a little shove.

• By ‘little shove’, of course, I mean a huge shove to go from 8km/s to 10.9km/s.

Launch Economics

• To enter low-Earth orbit (LEO), a rocket has to reach a velocity of 8km/s (28,800km/h).

• It costs ~$10,000 to put 1 kg into LEO.• Cost hasn’t changed much since the

1950s, because we’re still using the same technology – chemical rockets.

Launch Economics

• You have to time your launch carefully.• Earth revolves around the Sun every 365

days, Mars takes 687 days.• So about every 2.1 years we make a close

approach to Mars. Brahe knew that, and after last week’s lab, so do you.

Geometry of the journey