In a recent episode of MythBusters, the crew looked at the common movie scene where a rocket propelled grenade hit some type of vehicle. Typically, the RPG explosion and impact will flip the car over in a most dramatic style.
The two images above show the results of this episode. When a real RPG hit an SUV, it just exploded but the vehicle pretty much stayed stationary (show on the left above). In order to get the car to flip on impact, they did some movie special effects (fuel for a bright explosion and a piston to push the car over). Really, you should check out the MythBuster tumblr page with all the cool gifs of the action.
So, why doesn't the RPG flip the vehicle? Let's do some rough calculations to see why this doesn't work. While we are at it, we can estimate what it would take to flip the car.
Conservation of Momentum
One of the things we like to do in physics is to make a problem as simple as possible. In the RPG-suv collision, the SUV didn't even really recoil backwards from the impact. Let's just look at a one dimensional collision. If a vehicle doesn't recoil back, it surely isn't going to flip over.
Now, to look at conservation of momentum. What does that even mean? First, a definition for momentum. At these speeds (not near the speed of light), the momentum is simply the product of mass and velocity. There is a probably a historical reason, but physicists use the symbol p for momentum.
The momentum principle says that the nature of a force is to change the momentum. Over some time interval, this is written as:
Now consider two objects colliding. Since forces are interactions between two objects, the two objects exert the same force on each other (but in opposite directions). Since the collision last the same time for each object, the two objects would have opposite changes in momentum. Typically, we like to write this to say that the momentum before the collision is the same as the momentum after the collision (conservation of momentum).
Here is the RPG and truck both before and after the collision.
I will assume that the RPG sticks inside the truck during the collision. Yes, I know this is actually wrong but it will give a good value to start with. Technically, the momentum isn't conserved either since the RPG has a rocket on it (that's what the R stands for in RPG).
Setting the before and after momentum equal, I can solve for the final speed of the SUV.
Now I just need some values for the masses and the RPG velocity. I usually just get an estimate for some of these things, but I wasn't sure about the RPG speed. This Wikipedia page puts the speed at around 115 m/s with a mass around 3 kg. For the SUV, it looks like a Chevy Suburban that the MythBusters used. This has a mass around 2,500 kg.
Putting these values in, I get a final RPG-SUV speed of just 0.14 m/s. That's not very fast, but maybe you know that.
How could you get a larger SUV recoil? There are two things you could change. Make a heavier RPG or make it go faster. Before I change these, how fast would it need to recoil to get the effect we want? Here is a very rough estimate. Suppose that the rocket hits the SUV from underneath such that the vehicle would recoil upwards. If I want it to get the car to rise 2 meters, we can treat this just like a projectile motion problem. Using one of the kinematic equations (and knowing at the highest point the velocity is zero):
Putting the numbers in, I get a required recoil speed of 6.2 m/s. So, let's say I want to increase the mass of the projectile to get this speed. In that case I can rearrange and solve for the value of m1:
The required mass would be 142 kg. That's not going to happen. Ok, then what about increasing the speed? Here is the required speed of the RPG:
Just for fun, let's say the projectile has a mass of 10 kg. The required RPG speed would be 1156 m/s. That's pretty fast also.
You are not going to flip an SUV with an RPG this way. Notice that a real RPG would likely hit at an angle and not straight underneath the vehicle.
Conservation of Energy
Oh, I know you are already complaining. What about the explosion? What about that? Yes, the "G" in RPG stands for grenade and that means there is an explosion. However, if the SUV goes up there still has to be something that goes down to make momentum still conserved.
Suppose the RPG hits the engine and the engine breaks into two pieces with one piece going straight down. This would cause the rest of the SUV to go up.
Suppose that the two parts of the engine each have a mass of 150 kg. I can use a similar expression as above to determine the downward speed of the engine part. The difference is that I will assume the momentum of the RPG is negligible such that the total initial vertical momentum is zero.
I changed the subscript to e for the part of the engine that is shot down and s for the SUV (or the stuff left after part of the engine exploded out). Using the same recoil speed of 6.2 m/s (this is just an estimate) I get an engine downward speed of 97 m/s. That's pretty high but not impossible.
Now for the energy part. Both of these pieces had zero kinetic energy before the explosion but then they were moving afterwards. Where does this kinetic energy come from? It comes from the energy in the explosion. But how much energy is this? I know both the masses and the velocities of the two objects so I can calculate the final kinetic energy.
Using the values from above, I get a total final kinetic energy of 7.5 x 105 Joules. If all of this energy was in the RPG warhead with a mass of 3 kg, it would have an energy density of 2.5 x 105 J/kg. Wikipedia lists the energy density for a couple of explosives. TNT has an energy density of 4.6 x 106 J/kg. The RPG probably isn't TNT loaded, but at least it seems possible.
So, there could be enough energy to flip a car - but only if all of the ejected parts from the explosion are straight down. Of course, RPG's aren't really designed to flip trucks over. Instead, they are built with a warhead that can penetrate tank armor.
