Most people see this video of the SpaceX Grasshopper on a test flight and point to the funny stampeding cows. Yes, that's funny but that's not what I see. Instead, I see a great video example that can be used as a video analysis problem. Yes, the cows were funny too.
Video Analysis
This is pretty much a perfect video to analyze. The camera seems far enough away from the action and it doesn't pan or zoom. Also, I can use the length of the rocket to scale the video. I prefer to use the free and awesome Tracker Video Analysis, but of course there are other options.
I assume this is the version 1.1 of the grasshopper with a height of 160 feet. With that, I get the following plot for the motion (trajectory) of the top of the spacecraft.
A trajectory like this isn't actually very useful (but it does show the path of the rocket). But what about the forces? The rocket engines can provide both a vertical and a horizontal thrust on the craft. Here is a force diagram at some particular instant during the flight.
Here I am assuming that the thrusters point in the direction the rocket is facing - but this isn't exactly true. Remember, this is just an estimation. If I break these forces into x- and y-components, I can relate these to the horizontal and vertical accelerations:
I can get values for the accelerations and the angle. It might seem that I could eliminate the FT term and solve for the mass. However, this is what you would get.
The thrust is the only force without a mass term in it. When you eliminate the thrust, every term as a mass and they cancel. Bummer.
Well, I can still look at the acceleration. Here is a plot of the vertical velocity.
The rocket accelerates in the positive direction and increases its velocity. The slope of the velocity graph during this time gives a vertical acceleration of 1.8 m/s2. After that first part, the rocket slows down to so that it can stop rising. The acceleration during this second part is -2.71 m/s2.
What about the horizontal acceleration? Here is a plot of the horizontal velocity. Just a note. You can get a bunch of data points using Tracker Video's autotracker. Check that out for sure. Second, I am really marking the position of the rocket, not the velocity. To get the velocity, the Tracker program looks at a few of the position points to determine the velocity. This is why the velocity graph isn't so smooth.
Using the slope to find the two accelerations here, I get -1.8 m/s2 and 1.15 m/s2.
Homework
I normally save homework for the end, but I am making an exception here. For these questions, you can assume that all forces on the Grasshopper act at the center of mass (which they clearly don't).
- Find or estimate the mass of the rocket. What is the variation in the thrust force based on this motion?
- Assuming the mass and thrust remain constant, how long would it take this rocket to reach mach 1 (you can ignore air resistance if you like).
- Estimate the rate that fuel is used based on the mass of the rocket. How long could it hover in place?
Is there a way to estimate the mass of the rocket based on this video? I don't think there is, but I could be wrong.
How Does a Rocket Accelerate Horizontally?
In the above force diagram, I pretended like the rocket was just a point mass. Of course it isn't a point, it's more like a cylinder. Here is a better diagram of the rocket as it accelerates to the left (using an exaggerated tilt).
But why doesn't the rocket "fall over" in this orientation? If you look at the bottom of the rocket, it's clear that the gravitational force (acting at the center of mass) would exert a torque and make it rotate. Right? If you take a pencil and hold it up from the bottom just like this rocket, the pencil will indeed fall over. However, this rocket is accelerating horizontally also. The easiest way to deal with this acceleration is with a fake force.
What is a fake force? Well, forces are an interaction between two objects. I like to call these real forces. For example the gravitational force on the rocket is an interaction between the Earth and the rocket. If you just use real forces, then I can write the following:
Using this, everyone in the entire universe would be happy. They would be happy as long as the acceleration (and the forces) are measured from a non-accelerating reference frame. However, what if you want to use this awesome force equation for the case where reference frame is accelerating? What if I want to accelerate right alongside of this rocket to see if it falls over? In that case, I would be an accelerating reference frame and I would have to add a fake force to make everything work. The fake force would be:
In the above example, this fake force would be to the right and the net torque would be zero. The rocket would not fall over.
Fake forces aren't real, but you use them all the time. I know you do. When you are in your car going around a turn to the left, you feel like there is some force pushing you to the right. There isn't. This is just you pretending there is a fake force because you are inside an accelerating car.
How Do You Tilt a Rocket?
Ok, so you have to tilt this rocket to make it horizontally accelerate. I don't know much about the Grasshopper, but it seems like the only thrusters are the main rockets at the bottom. Could you just point the thrusters to the left to make it accelerate to the left? Nope. Here's a diagram showing a rocket initially at rest with angled thrusters.
What would happen to this rocket? Yes, you are correct. The rocket would start to tilt to the right due to the torque from the thrusters. That's not so useful, is it? Then how do you get the rocket to have a horizontal acceleration without tipping over? Now, I'm not a rocket scientist but I would guess that you have to first point the thruster to the right to get the rocket to tip to the left and then you can accelerate to the left. Here's a diagram showing what I mean.
If there were tiny thrusters at the top of the rocket, you could use those to tilt the spacecraft. But without those, I think this is what you have to do.
Another Tilting Example
What if I don't have a rocket, but instead I have a rod. Also, instead of a rocket thruster what if I push this rod with my hand? If you are standing still holding up a rod (supported from the bottom), how do you accelerate it to the side? Let's find out.
Here is a video I made just for you.
Hopefully you can see the motion of my hand moving first moving in the opposite direction as the intended acceleration of the rod.
Bonus Video
Here is a different rocket launch (a Cruise Missile).
Do you notice anything different about this cruise missile compared to the Grasshopper rocket?
