Chapter 1: Forces of Attraction
Hearing the ringing sound, Professor Dirac began searching for his phone which he suspected lay hidden on his desk. The desk in question was particularly nice – handcrafted from beautiful oak, with ornate brass handles on the drawers. Not that Doctor Pauli could tell this, however. At present, the desk was strewn with papers, scattered with pens, and littered with the debris of various other stationary. Such were the conditions that Dirac chose to work in. Papers began flying in various directions as he was still struggling to locate the source of the ringing. With a chuckle, Doctor Pauli stood up from his chair and reached over to a nearby shelf.
“Is this what you’re looking for?” He asked, passing the phone to Dirac. The professor quickly took the phone.
“Hello, this is Professor Dirac. How can I help you?”
“Aha, we’ve managed to reach you!” Exclaimed a booming voice from the other end of the phone. “This is Mr Planck, from the government. We have a task that may be of interest to you. A new planet has just been discovered by our top astronomers, and we suspect it may be habitable. We’re creating a team to travel to the planet – would you be willing to be a member?”
Dirac pondered this. It was a tough question. On the one hand, he would get to go to space! Everyone dreamed of going to space as a child, and he finally had the chance to experience something so few others had. On the other hand, he did not trust himself to be safe in space. What if he opened one of the windows? Or perhaps, he thought to himself, they wouldn’t even have windows. But then how – he brought himself back to the present.
“I will go, as long as Doctor Pauli will be allowed to accompany me.” He said finally.
Mr Planck laughed – given how notoriously forgetful Dirac was, it was reassuring that he always had Pauli to rely on.
“Of course. Now I’ll talk to you both later, but right now I must get back to organising this space mission. It’s not as easy as it sounds!” said Mr Planck, hanging up.
It was only a matter of weeks before the launch day arrived. “Are you nervous?” asked Dirac. Pauli contemplated his response. The two men were sat on a park bench, admiring the glistening lake before them. The sun was shining, and the trees around them rustled in the gentle breeze. It was a lovely day for a walk – they would not, however, be going on a walk. Instead, in barely an hour the two men would be soaring through space.
“A little,” came the reply, “I was just thinking about how the spacecraft will take off. When the spacecraft is on the ground, it has something to push off from. But what about when we’re in space?” Pauli had a hint of worry in his voice, but undoubtedly the question was mainly his curiosity.
“Now that is a good question,” smiled Dirac, “I suppose the rocket must not be pushing off of anything.”
“But then how could it move?”
Dirac thought for a moment, trying to remember that book he read 5 years ago – or was it 6 now? It really was a good book. Maybe he should read it again. After all, one can forget so much in just a few years… He snapped out of his aimless train of thoughts, suddenly remembering the answer. “I’ve got it!” The professor exclaimed. “When burning fuel, mass is launched out of the back of the spacecraft. And since you’re shooting mass downwards, you are pushed upwards!”
Both men sat thinking for a moment.
Pauli spoke first, “That does make sense. I wonder if we could find how quickly you accelerate upwards?”
“I think this is the equation” replied Dirac, writing something on the ground:
“α is the mass of fuel we shoot out every second, w is the speed of the fuel, m is the mass of the rocket and g is gravity. Oh, and a is the acceleration of the rocket.”
“Hmmm, I know that the fuel comes out at around 5000 m/s, the mass of the spaceship is 75,000 kg and that gravity is approximately 10 m/s2, so how quickly must fuel be shot out for the rocket to take off? And I wonder if that would be different on a bigger planet?” Asked Pauli. “I also wonder if we could work out what mass of fuel would be needed to accelerate the rocket from rest, to a velocity of 10,000 m/s in 600 seconds, with the same values as before for rocket mass, gravity and fuel speed.”
Dirac pulled out a crumpled sheet of paper from his pocket, almost certainly containing some important information. He quickly scribbled his answers over it.
Scroll down for the solution!
“Well for the rocket to take off, we know that the acceleration must be greater than zero. So we first set a to zero, then rearrange to get
and then again to get
Putting in the numbers you gave (10 * 75,000/5000) would mean that the fuel must come out at a rate of at least 150 kg/s. And if a planet is bigger, it must also have more gravity, as gravity depends on how massive something is. So that would mean that we would need to burn more fuel just to take off!” answered Dirac.
“The last question is a bit harder though. We first need to work out the average acceleration, by using the definition of average acceleration:
where v is the final velocity – so 10,000m/s – and u the starting velocity, which is zero here. Using this formula we can find the acceleration (10,000/600 = 16.7 m/s2), and this allows us to find the rate of fuel ejection as before. We get:
Putting in the numbers this is 400kg. Now that we have α, we can multiply by t = 600 s to get the total mass of fuel we need, m. Here, this gives us a value of 240,000kg of fuel.”
[…] Chapter 1: Escape From Earth with Professor Dirac and Doctor Pauli (I) […]
[…] Chapter 1 […]
[…] Chapter 1 […]
[…] House Magazine: Maths Challenge Escape From Earth with Professor Dirac and Doctor Pauli (I) Envelope Packing […]