Forces and Motion

Acceleration

Now Try This!

Ion propulsion may be the best way to send future space probes to far away planets. (An ion is an atom or molecule that has lost or gained electrons, so the number of electrons no longer equals the number of protons and the atom is no longer electrically neutral.)

Ion drives ionize inert gas (usually xenon) and accelerate the resulting ions to very high speeds (e.g. 30 km/s) out the back end of the rocket. This then accelerates the spacecraft in the opposite direction. But ion propulsion is very gentle. Although it can accelerate a spacecraft to perhaps 10 times the velocity of a more traditional chemical rocket engine, the acceleration it imparts is very low so that it takes a long time to reach these velocities.

The Deep Space 1 spacecraft was launched in 1998 solely to test ion propulsion technology. Instead of firing a ferocious rocket blast for a few minutes as a normal spacecraft does, DS-1 used a miniscule continuous thrust for 678 days and attained a final velocity of 4.3 km/s.

1 day = 86,400s

1 year = 31,536,000s

Can you solve this?

Assuming its initial velocity was 0 km/s, what was DS1’s average acceleration? Express your answer in m/s2.

m/s2

You are asked to find the average acceleration, a, and distance travelled, d.

You know:

Time, t = 678 d
Final velocity, vf = 4.3 km/s
Initial velocity, vi = 0 km/s

You need to find equations that include these known variables.

$a&space;=&space;\frac{v}{t}&space;=&space;\frac{4.3&space;km/s}{678d}$

But the answer needs to be in m/s2 so the km/s has to be multiplied by 103 to convert to m/s.

And the 678 d has to be in seconds. Note that 1 d = 8.64 x 104 s, so:

a = (4.3 km/s x 103 m/km) / (678 d *8.64 x 104 s/d)

= (4.3 x 103 m/s) / (5.9 x 107 s) = 7.3 x 10-5 m/s2

This is a really low rate of acceleration but it happened every second for 678 days.

Or this?

How many kilometers did it go during its 678 day mission?

km

Now, to finish up by calculating distance, d, just remember Eqn 1.7:

d = ½at2 = ½ * 7.3 x 10-5 m/s2 * (5.9 x 107 s)2

= 1.3 x 1011m = 1.3 x 1011m x 10-3 km/m = 1.3 x 108 km

Looking at this page full of math it looks hard, but once again, this was only cumbersome because of the need to change units. Physics would be easier if we in America routinely expressed units in meters and seconds. Of course, then you’d have to remember that you are 1.75 m tall and 5 x 108 seconds old!

Ion drive spacecraft are OK for robotic missions, but it would be nearly impossible to carry enough food for astronauts to survive an ion-drive mission even to the Moon.