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What Goes Up...

What does it take to get into Orbit?

If you think for a moment about any of the satellites that orbit the Earth you will find that they all have two things (and probably only two things) in common. Firstly, they are all at a high altitude (certainly above the bulk of the atmosphere) and secondly, they are all moving round the Earth. If you think a little longer, you might realise that both these things require energy.

Getting Up There…

If you climb a set of stairs or lift a bag of shopping on to the kitchen table you need to put in energy to overcome the force of gravity. In other words you need to give the object gravitational potential energy. The amount of energy you need to give it depends on three things: its mass, the height it is lifted through and the strength of gravity. These all come together in the equation:

Gravitational Energy (Joules) = mass (kg) x gravity (Newtons/kg) x height raised (m)

So if you have a 2 kg chicken in the shopping bag and the table is 1.1 m high and gravity is (about) 10 Newtons/kg then the amount of energy you need to put in is:

Energy (Joules)    = 2 x 10 x 1.1
   = 22 Joules

If your mass is 30 kg and you have to climb 2.5 m to get to the top of the stairs then the amount of energy you need is:

Energy (Joules)    = 30 x 10 x 2.5
   = 750 Joules

Just for the record, you would need to climb the stairs nearly 600 times to burn off the same amount of energy as is in one small Chocolate bar (the Chocolate bar is not a standard unit of energy).

To make matters a bit more complex the strength of gravity (actually 9.81 N/kg at ground level) gets weaker as you go higher. However even at 400 km, where the International Space Station orbits, it has only dropped to 8.89 N/kg. For all applications on Earth, you can stick with 9.81 (normally rounded up to 10) N/kg. For the satellite questions at the end of this sheet, there is a spreadsheet to help you. To give you an idea though, to carry 1 kg of supplies to the International Space Station takes 3.69 million Joules, just to raise it that high, and the job is still only part done.

…And Staying There.

If you did carry the 1 kg of supplies, up a very tall ladder, 400 km to the International Space Station you would still be in for a little surprise as the astronauts and their temporary home whizzed past you at 27,600 km/hr. That's 7.7 km per second! If you then dropped your supplies in shock, they would start to fall back to the Earth, 400 km below. It is very important at this point to keep a tight hold of the ladder and climb back down, slowly.

How do you stop your supplies from falling back to the Earth? Let’s do a thought experiment…

If you drop the supplies they will fall to Earth pretty much directly below your feet. If on the other hand you throw them in the same direction as the space station they will follow a curved path and fall further away. The harder you throw them the further away they fall. You could even make them land on the other side of the Earth! Now, if you throw them at a high enough speed the Earth’s gravity will make them fall in a curved path that eventually brings it straight back to you. They never hit the ground! You may not be surprised to find out that at 400 km this special speed is 27,600 km/hr, exactly the same as the space station!

What we have shown is that to stay in orbit an object must also be moving fast enough. This speed varies with altitude, the higher you go up the slower the speed is. We have already seen that at 400 km it is 27,600 km/hr, down at 100 km it is not much higher; 28,200 km/hr. For the Moon, over 375,000 km away, it has to orbit the Earth at only 3,700 km/hr.

The important thing as that to move quickly needs energy; kinetic energy! The high speeds that satellites move at need a great deal of kinetic energy. The amount of energy is given by the equation:

         Kinetic Energy (Joules)   = ½ x mass (kg) x [velocity (m/s)]2

If you run in a 100 m race and finish in 20 s your speed is 5 m/s (on average). If your mass is still 30 kg then your kinetic energy is:

Energy     = ½ x 30 x 52
   = ½ x 30 x 25
   = 375 J

Take the example of a car with a mass of 750 kg moving at 17 m/s (a little over 40 mph), its kinetic energy is:

Energy     = ½ x 750 x 17 2
   = ½ x 750 x 289
   = 108,000 J

If the same car were only moving at 30 mph (12.5 m/s) its energy would be reduced to:

Energy     = ½ x 750 x 12.5 2
   = ½ x 750 x 156
   = 59,000 J

That is why far fewer children are killed in car crashes at 30 mph than 40 mph.

The speed that the International Space Station moves at (27,600 km/hr) is 7670 m/s. At this speed every kg of the station needs 29 million Joules. Fortunately, if you are on the equator you are already moving at 464 m/s, due to the rotation of the Earth. This means that you already have 108 thousand Joules of kinetic energy for every kg. Europe launches its Arianne 5 rockets from French Guiana, 5° north of the equator; the Americans have to launch their rockets from Florida, nearly 30° north of the equator. This gives the European launches an extra little kinetic energy to start with and so helps keep them competitive. Even so you will still need to give every kg of cargo a total of 32.6 million joules of energy and this needs to come from the energy released from burning rocket fuel.

Question 1


Image: Courtesy of Scaled Composites. LLC

On October 4th 2004 SpaceShipOne won the 10 million dollar Ansari X-Prize by carrying its civilian pilot, Brian Binnie, to an altitude of 100 km twice in a fortnight. SpaceShipOne’s mass was 1200 kg (without fuel). How much energy did the rocket motor need to release in order to get it to 100 km?

SpaceShipOne climbed to an altitude of 100 km and then came straight back down. Was it a satellite?

Question 2

To go in to orbit at 100 km altitude SpaceShipOne would need to be travelling at 7,840 m/s. How much kinetic energy would it need to travel this fast? How many times larger is this than the energy that it was given? How close to being a satellite was it?

 


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