Hi,
I was just wondering - why does a graph plotting g-forces against time in a spacecraft have a non-linear relationship?
Thanks.
Hey!
So, when plotting g-force against time, we have to think about what is actually happening to the shuttle. So, first let's look at the formula for g-force.
Where
g is the force due to gravity, and
a is the acceleration of the rocket.
So, since
g is approximately constant in the initial stages of the launch (it will get weaker as the rocket moves away from the earth, but let's ignore this as it will hardly contribute to the graph, at least for small
t), what we really want to know is the values for
a. If a increases linearly, so will the g-force. If a decreases linearly, so will the g-force. And, if a increases non-linearly (as we will soon find that it does), so will the g-force.
Now, we know from Newton's laws that
This is how we figure out what the acceleration of the rocket is. The burning of fuel creates a fairly constant force downwards and, by principles of conservation of momentum, the rocket is therefore propelled upwards. However, let's think about what's actually happening. As the fuel is being burnt, resulting in a constant force (ie. the left hand side of the Newton's equation is constant), the mass of the rocket is decreasing by quite a lot. This is because most of the mass of most rockets is made up of fuel! So, over time, the mass is decreasing. However, if the left hand side of the equation is constant over time, the right hand side must be as well! So, if the mass is going down, the acceleration must be going up. In other words, acceleration
increases over time.
Huh. Except, when you reach this point, you expect acceleration to increase linearly over time. Which means, as I've described above, that the g-force should increase linearly over time.
Damn.
Okay, so turns out I was wrong, but I'm going to leave my answer above for completeness, and to show the logical steps you should be taking (and, presumably, you did take, in order to reach the linear conclusion).
In that case, it is likely because g decreases as t increases (the rocket gets further from the earth). This will result in a non-linear relationship. Also turns out to be a way easier solution! Sorry for the rambliness but hopefully this all made sense; really great question!
Jake