The internet loves videos of motorcyclists doing crazy things on the autobahn. This one seems especially popular. From what I can gather, the rider is hauling along when a car cuts him off, requiring some sudden emergency braking. Let me show you how out of control I can get with the physics in this video. And don’t worry—I’ve got a homework question for you at the end.
Braking Acceleration
Of course, the first question that comes to mind is what was the acceleration magnitude during this event? I will start with the definition of acceleration (in one dimension):
To determine the acceleration magnitude, I simply need the change in velocity and the change in time. Easy. Watch the video and you’ll notice that at one point the speedometer reads 213 kph (kilometers per hour). After 1.968 seconds, it reads 124 kph. (Yes, I essentially counted frames to determine the change in time.) This means the change in horizontal velocity is 89 kph, or 24.7 m/s. Using the change in velocity and change in time, I get an acceleration of 12.55 m/s2, or 1.3 g’s. That’s pretty significant braking. I would even call it emergency braking.
That calculation wasn’t too complicated. In fact, it’s merely the average acceleration over that time period. Can I get a more detailed view of the bike’s speed? Yes, of course. This bike feature two important data displays. The first, obviously, is the digital speedometer, but I don’t think it displays the speed instantaneously. The second display is that big analog tachometer. It showed the revolution speed of the engine, in revolutions per minute (with a scale of 1,000). If the rider remains in the same gear and doesn’t engage the clutch, the engine rotation speed should be proportional to the motorcycle speed.
I can get a more accurate measurement of the speed by looking at the angular position of the tachometer needle with my favorite video analysis tool, Tracker Video Analysis. Here is the data during the breaking emergency:
Notice that the rate at which the angle changes appears fairly constant. This indicates that the motorcycle speed changes at a constant rate and that the acceleration is nearly constant. If you wanted to, you could use the exact rpm values in the video and their correlation with the speedometer to get a conversion from rpm to kph. Also, notice that the rider didn’t engage the clutch as he slowed down. If he had, you’d see the tachometer needle drop as the rpms fall.
Coefficient of Friction Between Tire and Road
Why does the motorcycle slow down? Don’t say “Because the rider grabbed a big handful of brake.” OK, that’s not technically wrong, but from speaking as a physicist, it’s best to say increased friction between the tires and the road is why the motorcycle slowed down. Although friction is a fairly complicated interaction between two surfaces, you can usually model the maximum magnitude of the frictional force with this equation:
In this expression, μk represents the coefficient of static friction. Its value depends on the two types of surfaces interacting—in this case, rubber and asphalt (or concrete). The friction force also depends on the force that the two surfaces are pushed together. Physicists call this the normal force since it is perpendicular to the surface (the ground). Here, the force N is simply the weight of the motorcycle, which you determine by multiplying the mass (in kilograms) and the gravitational field (9.8 N/kg).
For the forces in the horizontal direction, only frictional force pushes in the opposite direction as the motion of the motorcycle. Since there is only one force, it must be equal to the product of the mass and acceleration. Combining this with the model of friction and my expression for the force N, I can solve for the coefficient of friction:
Since I already know the acceleration in g’s, this calculation is easy. I get a coefficient of friction with a value of 1.3 (no units). Typically, this coefficient is between 0 and 1, but this is just a rough model. And I doubt the motorcycle could have slowed down at a greater rate than seen in the video without the wheels slipping, which would have been bad. Very bad.
Homework
I promised you homework, so here you go. Find the velocity of the car that pulls out in front of the motorcycle. Yes, this can be a tricky question. You could estimate the distances between the motorcycle and the car to determine a rough value. Or you could use the final speed of the motorcycle just before it narrowly avoids a collision. Or you could try a third method.
You can use the angular size of the car in each frame to estimate the distance between the motorcycle and the car. A few things make this easier, like knowing the angular field of view of the dash camera or the exact size of the car (both of which you probably could figure out). However, if you want to do it the hard way, you could use the change in angular size of the car along with the acceleration of the motorcycle.
Like I said, this can be challenging. Just to help, here is an example where I do something like that.
