I’ll be honest: I love my drone. I mean, I always had remote-control vehicles back when I was a teenager. And of course the most impressive RC vehicle was the gas-powered helicopter. But it was expensive and hard to fly. Now, with a quadcopter, it’s a breeze. On top of that, it takes pictures and videos.

Since I have this fascination with drones, it’s only logical to take the next step and use it for some physics. How about an analysis of the aeronautics of this particular drone, the DJI Spark. Drones, physics—what could be better?

So I used my phone to record some slow-motion videos of the Spark moving first vertically and then horizontally. Here’s an example below. And then I used one of my favorite tools, the Tracker video-analysis app, to plot the position of the drone in each frame. Armed with that data, it’s just a hop, skip, and a jump to derive performance specs like acceleration and thrust.

On the Ball

The video essentially gives me a series of time-stamped snapshots of the drone as it moves, but I need to know the frame rate to calibrate the time scale. My phone says it records slo-mo at 240 frames per second—or, in other words, at 4.17 millisecond intervals.

Just to double-check that, I’m going to run a test analysis on something I already know about: the acceleration of a ball tossed straight up into the air. An object in free fall, where gravity is the only force working on it, has a vertical acceleration of about –9.81 meters/second^{2}.

So if I put a meter stick in the video frame (it’s that horizontal stick next to my hand), I will know both the distance scale and the vertical acceleration. From that, I can figure out the true frame rate. Here’s what the ball toss looks like:

I ran the Tracker software on this clip and adjusted the listed frame rate until the fitting equation gives me a vertical acceleration of –9.81 m/s^{2}. After playing around a bit, I got a time interval of 4.28 milliseconds—so actually about 234 frames per second. Here is the trajectory with the adjusted frame rate:

The position of an accelerating object depends on both time and the square of the time. If you’ve ever taken an introductory physics course, you’ve seen this famous kinematic equation:

Fitting a quadratic function to this data shows that the coefficient in front of the t^{2} term should be equal to the acceleration divided by two. This gives an acceleration of –9.822 m/s^{2}. That’s pretty close, so let’s stick with the derived frame rate of 234 fps. Now, back to the drone!

Vertical Acceleration

I’ll start with the simplest case for acceleration—straight up. Plopping the video into the software program, I get this plot of position (height in meters) as a function of time (in seconds):

You can see that the drone starts at rest and moves upward with an acceleration of 4.755 m/s^{2}. But it doesn’t keep accelerating—before 1.5 seconds are up, it reaches a constant upward velocity of 3.67 m/s. That’s why the line becomes straight. Looking at the telemetry data from the drone controller, it gives an upward velocity of around 3 m/s. So this all seems fine.

Now for the fun part. What is the thrust force from the drone’s four rotors? First, if I know the mass (**m**) of the drone and its vertical acceleration (**a _{y}**), we can find the effective

*net*force in the vertical direction with this force-motion relationship:

That net force, in turn, can be decomposed into two distinct vertical forces: (1) the upward thrust force, **F _{T}**, and (2) the downward gravitational force,

**mg**—what you call “weight” in everyday life, which is mass times the local gravitational field

**g**(9.8 N/kg). So

**F**=

_{net-y}**F**–

_{T}**mg**. Subbing that in and rearranging, we get this expression for the thrust force:

We know all of those values. From the video analysis, remember, **a _{y}** = 4.755 m/s

^{2}. DJI’s spec sheet lists the weight of the drone (

**mg**) as 0.3 kg. Putting this all together, I get a thrust force of 4.37 newtons. Cool!

Oh, what about air resistance? Well, at first the drone is moving very slow, so the air resistance would be negligible. But once it reaches a constant upward speed, that could be a factor. At a constant speed, acceleration is zero, but now there are three forces acting on the drone: the upward thrust force and the downward forces of gravity and air resistance. We could estimate the drag coefficient in this case, but I’ll leave that as a homework question for you.

Forward Acceleration

Now let’s look at the drone as it accelerates horizontally. Here’s a plot of horizontal position as a function of time:

This time I get an acceleration of 4.88 m/s^{2}. Notice that in this case the drone accelerates during the entire time observed, which is just over 1.5 seconds. Of course, if we watched it for longer, it would reach some constant velocity.

But what about the value of the acceleration? It’s pretty close to the vertical acceleration of 4.755 m/s^{2}. Shouldn’t it be much higher, since it doesn’t have the gravitational force pulling it down? You might think that, but the drone thrusters still have to deal with the downward gravitational force in order to keep the drone from falling.

Here’s a close-up of the drone as it begins to accelerate forward. There’s some important stuff here—and it looks cool.

This is what makes the quadcopter design so awesome. You can move in any direction—up, down, forward, backward, sideways, diagonally—and all of these moves are just changes in power to the four motors. The only moving parts are the four horizontal rotors. You don’t need any complicated pitching blades like in a helicopter.

In this case, the front rotors decrease in power (and thus produce lower thrust), which causes the drone to tilt forward. At this point, the thrust from all the rotors is up and at an angle. This provides a component of thrust in the forward direction that accelerates the drone. Here, this force diagram might help:

But now I have two methods to calculate the total magnitude of the thrust. First, I can look at the vertical forces. In this case, the net force in the vertical direction would be zero, since the drone doesn’t accelerate up or down. The downward gravitational force and the vertical component of thrust exactly offset.

Or I can approach it from the horizontal direction. Here, the net horizontal force should be the product of the mass and the horizontal acceleration. These forces are vectors, which means they can act in both the horizontal (*x*) and vertical (*y*) directions at the same time. However, since the *x* and *y* directions are perpendicular to each other, the components of these vectors form a right triangle. Trig functions are really just ratios of the sides of right triangles. Boom—I can now find the components of thrust in the *x* and *y* directions.

With a tilt angle, θ, of 33.8 degrees (measured from the video), these equations give me two different values for the total thrust magnitude: Working from the vertical forces, I get 3.54 newtons. Using the horizontal forces, I get 2.63. But wait! Why are these different? And why is the vertical result different from what we got before, when the drone was moving straight up?

First, for the difference between the two calculations: It’s possible that there could be a significant air resistance pushing against the horizontal acceleration. If the air resistance has a horizontal magnitude of 0.504 newtons, then both of these methods would produce the same magnitude of thrust.

As for why the vertical result, 3.54 newtons, differs from what we got when the drone was going straight up (4.37 newtons): Well, they aren’t *that* different—it could just be measurement error. But there is another plausible explanation. Maybe the drone software limits horizontal acceleration to some reasonable value? After all, most of the bad things you can crash into are in that direction.

OK, I have one more homework question for you. Measure the horizontal acceleration for the drone as it slows to a stop. I suspect it will be greater than the acceleration for speeding up, since the air-resistance force will be pushing in the same direction as the thrust. That’s my guess. Let me know if this is true!

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