A Motorcycle, a Car, a Jet: Which One Would Win?

I found this video after look at physicsvideos.net which was mentioned in this tweet:

Let me just point out that this might be the worst video ever in terms of analysis. These people had enough money to get a fighter jet race a car and a motorcycle, but apparently they couldn't afford a tripod or a decent video camera. I just don't get that. Just imagine how difficult it would be to reproduce this video now. I don't believe that any producer would easily be able to get a jet and car on the same runway. Too dangerous and too expensive.

Video Analysis

Let me start with a quick shot shortly after the race begins.

With the camera shaking around like it is, this isn't the easiest thing to analyze. Normally, I would try to mark some location on the ground and use that as the origin. However, for this case the low resolution (on top of the crappy video) this method won't work too well. Instead, I will try something different. I will use the jet plane as the origin. That way I can get the motion of the other two vehicles relative to this jet.

Ok, but what about the scale. After looking around for a bit, I think that this jet is a Mirage F1 - I could be wrong though. With this, I can set the length of the jet to 15 meters - again, that could be wrong.

Here is the position of the motorcycle and the car during this shot. I have also included a spot on the runway (once you can see it).

From this, it looks like all of these vehicles are moving at nearly a constant velocity (the acceleration must be small). But don't worry - I will fix that in a little bit. First, I need the velocity of the jet. Since the jet is the origin, it will have a velocity that is the negative of the ground's velocity. From the graph, it looks like the jet would be moving around 31 m/s (69.3 mph). Relative to the jet, the car has a speed of 3.92 m/s and the motorcycle is at 14.2 m/s. This would give them velocities relative to the ground at 34.92 m/s and 45.2 m/s.

But what about the acceleration? Well, without more data I can only look at the average acceleration. Let's assume that all of the vehicles start from rest at the same time and only move in the direction of the runway (which I am calling the x-direction). In this case, I can find the average acceleration as:

Since the scene from above was 2.9 seconds after the start of the race, I get the following average accelerations.

• Motorcycle = 15.6 m/s²
• Car = 12.04 m/s²
• Mirage F1 = 10.69 m/s²

But what about later in the video? At some point, the video once again shows a view of the vehicles that only partially sucks (but it doesn't completely suck). Here is that later data.

Now using the same method as the previous analysis, I get the following average velocities:

• Motorcycle = 69.1 m/s
• Car = 88.2 m/s
• Mirage F1 = 101 m/s

Ok, there are problems here. First, the data for the car is whacked out. As the jet gets farther down the runway, it looks smaller and has a different angle with respect to the camera. Since the video was terrible, I didn't even try to take these factors into account. That means that by the time the car comes into the frame, the scale is probably off. However, if you look at the speed of the car, it's around 195 mph - that seems too high, but the video says it's a Porsche.

What about the average acceleration? The second scene starts 8.8 seconds after the start of the first scene. If I use this as the change in time, I can use the change in velocities to find the average acceleration during this second interval. I get the following:

• Motorcycle = 6.08 m/s²
• Car = 8.65 m/s²
• Mirage F1 = 10.26 m/s²

Here is the key. The motorcycle's average acceleration is lower than it was before - same for the car. These two vehicles can't keep accelerating to reach super high speeds. The jet's acceleration is essentially the same as it was before. It just keeps getting faster and faster. Now, be careful - don't confuse acceleration and velocity (a common error). It's the velocity of the object that leads to it overtaking another object.

Let me list a couple of important times (needed for the homework below). If you watch the video, the jet is at the same position as motorcycle at a time of 21.5 seconds. The car passes the motorcycle at 25.6 m/s.

Modeling the Motion

The motion of the motorcycle and car aren't so simple. They clearly don't have a constant acceleration (or they would win the race). So, how do you model this race? One way is to just use a constant acceleration for both the car and the motorcycle until these vehicles reach some maximum speed. At that point, the acceleration could be zero.

You could do this with just plain kinematic equations (because the accelerations are piecewise constant) - but what fun is that? Here is another way to do it with a numerical calculation in GlowScript. Here is what that would look like if you ran it.

But really, you should play with the program yourself. It's not difficult. You can do it, I know you can.

I guess an animation is pretty cool, but maybe it's not that useful. Here is a different version of the program that includes a graph.

That graph shows the position of the threee vehicles. If you wanted, you could easily change the graph to display the x-velocities of the vehicles.

Homework

Don't act surprised. I told you I was going to give you homework.

• Sketch a graph of velocity vs. time for the three vehicles using the model that has the motorcycle and car with a constant acceleration until reaching some maximum velocity. Check your sketches by modifying the GlowScript program to plot velocity vs. time.
• Does the GlowScript model have the jet passing the motorcycle? at the correct time? What about the time for the car to pass the motorcycle? Can you adjust either the acceleration or maximum velocity so that the model agrees with the video?
• Create a different acceleration model. What if you add some velocity-dependent drag force. Would that give a better model for the motion of these vehicles in this race?