I hope you didn't miss the last MythBusters. Only the MythBusters have a big enough budget to launch their fourth rocket car (two in this episode). The above clip shows you all the great details. The part I find the most interesting was the view of the wooden ramp after the jump. Of course the rocket from the car tore it up a bit, but there were also big grooves where the tires from the car dug into the wood.
This leads to a simple question: what kind of force did the car exert on this ramp? Why does the car even push on the ramp with a force greater than the weight of the car? Let's look at the forces on the car during this jump.
The important thing in this diagram is the momentum (the dotted arrow with the vector p). The rocket on the back of the car clearly increases the magnitude of the momentum, but can it make the car turn up? No. Perhaps we should all recall the momentum principle as it applies in a case like this.
Over some time interval (Δt) the net force is equal to the change in momentum over the change in time. Both the net force and the change in momentum are vectors - so direction matters. Let's just consider the case where the car is moving at a constant speed. During the short time it moves on the ramp, the magnitude of the momentum wouldn't change. However there would indeed by a change in the vector momentum since the car changes direction.
If I look at the momentum vector at the beginning and end of this time interval, I can find the change in momentum and the net force.
The faster the car moves, the greater the momentum it will have. This also means that it will change directions in less time. A large force will be required from the ramp in order to get this change in momentum. Since the ramp pushes on the car, the car will have to push back on the ramp with the same magnitude (it is the same interaction). This is why the ramp breaks.
Video Analysis
How about an estimate for the magnitude of the force the ramp exerts on the car? Yes, let's do it. The video on the MythBusters site is a great place to start. They show a nice slow motion video. Before using the video to calculate the change in momentum, I first need to determine the frame rate and the scale.
The scale should be fairly easy since they used a known car - the Impala. I am pretty sure someone said they used the 1966 Chevy Impala. This site claims a length of 213.2 inches and a wheel base of 119 inches. If I use that, I can easily scale the video.
What about the time scale (frame rate)? This isn't as easy as it would seem. The best shot of the car going up the ramp is in slow motion with a stationary camera. However, I am fairly certain the frame rate in this video is not constant. Instead, I will use a slow motion and panning video. Comparing the time of the car on the ramp to a clip at normal speed, I think this clip is about 0.36 the speed of real life. I'm not certain, but I think this will be good enough. I tried to set the time scale based on the acceleration of projectile motion debris, but this didn't work. Well, it worked but I am pretty sure the first part of the video (before the debris) was at a different frame rate.
After adjusting the video frame rate (to make it the real frame rate), I get the following plots of the car as it goes onto the ramp. This first plot is the horizontal position.
Is the car accelerating? Maybe. Two things happen here. The velocity direction changes when the car hits the ramp, so there should be a decrease in the horizontal velocity. But then there is the rocket causing an acceleration. On top of these two things, there could be a perspective error. In the end, all I can really say is something about the average horizontal velocity. This would be about 32.8 m/s (85 mph). That seems about right.
Here is a plot of the vertical position.
Again, the speed after hitting the incline is close to constant. This puts the average vertical velocity (while on the ramp) at about 8.7 m/s (19.5 mph).
With this, I have can put together a before and after velocity vector. If I estimate the mass, I can also get initial and final momentum vectors. With the extra weighted bumper and the rockets, I am going to estimate a car mass of 2,500 kg. What about the time? That is the tough part. Let's look at the damage on the ramp from the tires.
From this, it looks like the tires were interacting with the ramp for about 3 meters. If the car was traveling at a speed around 33 m/s, this would give a contact time of about (3 m)/(33 m/s) = 0.09 seconds. I can now find the average net force on the car during this time. Here I am using my favorite vector representation.
The ramp essentially just pushes up. How does this force compare to the force for a car just sitting there at rest on a flat ramp? In this case, the ramp force would just be the weight of the car. For a 2,500 kg car this would be 2.45 x 104 Newtons. That is significantly less than the force for the rocket car. I guess you could say with the car moving at a high speed, the force on the ramp would be like 10 cars stacked on top of each other. And that is why the ramp failed.
Homework
This rocket car on the ramp is great for all sorts of homework questions. First, let me point out what would make this clip even better: frame rate listed on the video. Wouldn't that be great? As long as I am dreaming, how about listed dimensions for different structures (like the size of the ramp). And one more thing, easily downloadable videos.
Now for the homework.
- What is the thrust produced by these rockets? There are two approaches here. You could make this an internet scavenger hunt and try to find details on those particular rockets. The other option would be to look at the video and use data from that. If you look around at the different angles, I suspect you can find one that shows enough of the car accelerating so that you can get an estimate of the thrust force.
- Plot the trajectory of the car after it leaves the ramp. This might be another method to estimate the thrust of the rockets since the car accelerates upward at one point.
- Based on the rotation rate of the car after it leaves the ramp, estimate the torque and torque arm distance for the rocket about the car's center of mass.
- Now that you have an estimate for the thrust force, what is the terminal speed (on the ground) for this rocket car if it never hit the ramp?
Obviously, you will need to perform some video analysis to complete this homework.
