Built on Facts

Whew! Long weekend. Unfortunately, due to work/procrastination the post I wanted to write still isn’t ready. But today I’m also teaching 1d accelerated motion to my Physics 218 students, and that’s interesting of itself.

One of the things I try to do is give problems that help build instinct for what an answer should look like. If an accounting students calculates that a million-dollar-per-year business owes a billion dollars in taxes, he will instantly recognize that there’s an error somewhere. You’ve got to built up the same kind of intuition in physics. Example:

The New Horizons mission to Pluto travels at a constant(ish) speed of about ten miles per second and will take around nine years to reach its destination. If you could travel at a constant acceleration equal to the acceleration due to gravity at the earth’s surface (a = 9.8 m/s^2), how long would it take you to travel from Earth to Pluto, starting from rest? (Assume the distance to Pluto is 6×10^9 km)

Ignoring all the obvious effects and treating it as a 1-dimensional motion problem, give it a shot!

The equation for distance traveled under accelerated motion assuming a zero starting speed is:

Unfortunately chemical reactions can’t generate large constant acceleration for very long before the fuel runs out. Future advances such a nuclear propulsion, solar sails, and ion engines might help fix this.

Comments

#1 Eric
February 2, 2009

See if I can do this in my head.. a little typing to help out.
6×10^12 meters * 2 / 9.8 ~= 12×10^11
sqrt(1.2×10^12) ~= 1.1×10^6 seconds
11×10^5 / 8.64×10^4 ~= (serious estimation doing this in my head) 12.5 days? Maybe?

Calc check gives 12.8. Not bad. Unless of course I screwed both calculations up…
#2 dominich
February 2, 2009

Re #1: If you don’t mind going past at .03c then thats OK.

Adjusting for turnover and deceleration ends up with t=2*sqrt(d/a) which gives a shade over 18 days.
#3 Uncle Al
February 2, 2009

If an accounting students [sic] calculates that a million-dollar-per-year business owes a billion dollars in taxes, he will instantly recognize that there’s an error somewhere.

There’s no error anywhere – IRS Form 706, plus penalties and interest – though the kid is unemployable. A functional student would find a $billion in tax refunds, re NINJA loan and contingent derivative securities’ bailouts.
#4 Eric Lund
February 2, 2009

#2 is correct: you actually want to go to Pluto, not zip by at high speed. My in-my-head estimate gave ~9 days of acceleration and the same amount of time for deceleration. The calculator says 9 days 1 hour and change for each half, or just shy of 18 days 3 hours for the total (neglecting the time it takes to turn over and start decelerating).
#5 Karl Withakay
February 2, 2009

New Horizons is a fly by mission, so ~12.7days.

I suppose relativity could confuse things a little at a final velocity of 3% the speed of light as there will be some time dilation involved.
#6 Paul Johnson
February 2, 2009

I just finished the chapter on gravitation in the Morin book in my particle and wave dynamics class and we had a similar but seemingly more complicated problem.

If the earth stopped in its orbit and began to fall slowly into the sun due to the sun’s gravity how long would it take? It was a fun one because it taught us to look for clever physicist alternatives to solving differential equations.

What I want to know is that without a way to exceed c in 3-dimensional space is there an alternative to semi permanent space borne stasis? Assuming the earth will eventually go. And besides ridiculous things like wormholes…
#7 andy.s
February 2, 2009

I think I remember a Heinlein character doing this calculation while imprisoned by carnivorous aliens in a cell on Pluto. (Have Space Suit, Will Travel).

So do this problem for extra credit without pencil or paper while waiting to be messily devoured.

Don’t worry, though. In the end, the aliens got their comeuppance when their home planet got sent into another dimension (without their sun).
#8 Rick Pikul
February 3, 2009

Well, solar sails and ion/plasma drives can’t do 1g, and for a nuclear drive you would need staging or a better than probably possible nuclear salt-water drive.

The future advancements you want are fusion and/or anti-matter drives.

(12.7 days at 1g needs an exhaust velocity of 3500km/s to keep the mass ratio within the practical limit of 20.)

The good news is that existing designs are fine for an Earth-Pluto Hohman transfer, even chemical rockets can do it with a mass ratio under 3 and a solid-core nuclear thermal, (which have been around since the 1950′s), can do it with a mass ratio 0f less than 1.5.

(Yes, I’m a fan of Atomic Rockets at http://www.projectrho.com/rocket/ .)
#9 Rob
February 5, 2009

Any rocket would need to emit some kind of mass to drive it forward (except for a solar sail). To keep this “fuel” mass low, the velocity of the emitted mass should be as large as possible. NASA’s fusion mirror rocket prototype (which is too small to work by the way) would enable quite high mass emission velocities AND the energy to get the mass there. Unfortunately the fusion mirror is not perfect and would probably need a conventional fission reactor to supply it with energy.

But it’s a good idea, and it will probably work with some real technological effort put into it. The problem is as always: who would want to pay for one? Asteroid mining companies? Hafnium and Indium might pay for it if the amount returned isn’t too large. Traditional metals as gold and platinum could also help.
#10 David M. Freeman
April 8, 2009

Endless power for deep space exploration

Abstract:
Antimatter: Magnetic Plasma Ion Engine System:

Can be contained by a magnetic flux, within a timed expansion of the magnetic and gravitational fields in mil. seconds, within a given area of space. A spherical containment vessel is where this process will take place. The use of six magnetic rings around the containment area moving in opposite directions and orbiting the containment will create an artificial gravity field, which can be controlled by the use of a magnetic flux, with in the gravitational field around the antimatter within the center of the chamber. This will hold the process of heat generation in place around the antimatter core. Shielding around the containment will be used to protect the crew and other personal from both heat and radiation produced during the process. Plasma shielding is the main element in use for this level of protection and control. This Containment must be held within a perfect vacuum in order to allow heat generation to be effective in the creation of Ions at a given rate and speed with in the containment and within the injector tubes. These Ions will then be stored in a holding area developed to maintain their rate and speed. This vessel will need to handle temperatures of up to 2,000 degrees (K) with in the inter-containment area. Nitrogen will be used to cool the core of the chamber and keep the antimatter stable. This process will produce an endless supply on Ions for use in the Ships Ion engines. This use of antimatter will pave the way for sub-light engine development in the near future. The on broad computers will be used to control the rate and speed of the Ions being produced by using the flux controller to very the magnetic and gravitational fields with in the chambers core. By speeding up the Ions before entering the engine injectors, high speed can be meet with in minutes in stead of in hours or days. Most Ion engine designs take a long time, before the thrust will push the spaceship at a rate needed for inter planetary travel. This system will make it possible to change all that for the future. By using Plasma as the catalysis to produce Ions the possibilities are endless as far as speed is concerned. As long as you can maintain a stable environment with in the containment, sub-light speed is not out of the question.