Today, we're doing a challenge question

instead of a regular episode.

This question is going to draw on two episodes--

our interstellar travel episode from a few weeks ago

and next week's episode on killer asteroids.

We're going to use some rocket science to save the planet.

Here's the situation.

The near-Earth asteroid, Apophis,

is set to buzz by Earth on April 13, 2029 and then again exactly

seven years later on April 13, 2036.

It is currently predicted that it will miss both times,

but here's a hypothetical.

Something like a passing comet has perturbed it's orbit.

Apophis will still miss in 2029, but it

will hit Earth head on in 2036.

This 325-meter-diameter asteroid weights

around 30 billion kilograms, and it's traveling

at 30 kilometers per second.

It will be a devastating impact if we don't stop it.

You've been chosen to assess the possibility of deflecting

Apophis with a gravitational tractor.

For simplicity, it'll look like this.

Launch a spacecraft to intercept Apophis during its 2029 pass.

The spacecraft will hover just in front of the asteroid

and act as a gravitational tractor,

slowly increasing the asteroid speed

by the mutual gravitational attraction over the seven years

before the scheduled impact.

If we can pull Apophis just a couple of Earth diameters,

say 25,000 kilometers, ahead of its expected position

by the impact date, then it will miss us.

Question, assuming we have a state of the art

post-fusion rocket engine capable of exhaust velocities

of 500 kilometers per second ready by the launch data,

can we feasibly build a spacecraft

that would give the necessary bump to Apophis?

How massive would the spacecraft, including the fuel,

need to be?

To do this, you'll need lots of Newton.

Never mind Einstein for now.

This is the weak gravity regime.

You'll need Newton's law of universal gravitation

as well as basic mechanics and kinematics.

You'll also need the rocket equation

which relates velocity change to the propellant exhaust

velocity and the ratio of wet to dry, or fueled to unfueled,

mass.

One clue for now, you don't need calculus

to do this if you assume constant propellant

use over the seven years and calculate an average spacecraft

mass.

I'll release some clues over the "SpaceTime" Twitter

if you need help, but submit your answers to "PBS SpaceTime"

at gmail.com by November 20 and use the subject line, Spacetime

Killer Asteroid Challenge.

Use exactly this line, not case sensitive,

because we filter emails by subject line.

Now, please don't discuss or post answers in the comments

or, indeed, anywhere online.

Give everyone the chance to figure this out for themselves.

Good luck saving planet Earth.

We'll select five random correct answers

to receive PBS Digital Studios t-shirts,

and I'll have the solution for you in a couple of weeks.

Next week, we'll be back with a killer episode of "SpaceTime."

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