I must admit to being pretty excited about Wonder Woman. Although I read a lot of comic books when I was younger, I favored Marvel over DC, so everything I know about Wonder Woman came from the TV show starring Linda Carter. It sounds like Wonder Woman might be the best superhero movie so far.
Like all superhero movies, Wonder Woman provides an opportunity to do a little physics. Now, anyone who knows Wonder Woman knows one of her signature moves is deflecting bullets with her awesome Bracelets of Submission. Those things are really cool—they’re indestructible, and can absorb the impact of falls, bullets, and other stuff. But what does physics say about this? Let’s find out.
It’s All About Momentum
Momentum plays a big role in this discussion. In short, the momentum of an object is the product of its mass and velocity. Why would you multiply them? Because the net force on an object changes its momentum. Something else you need to know about the nature of force is that forces are an interaction between two objects. If object A pushes on object B, object B pushes back on object A with the same magnitude force.
Now suppose two objects collide. For now, I will use a golf ball and a basketball moving toward each other at the same speed. Just to be clear, the mass of the basketball is larger than the mass of the golf ball. When they collide, each exerts the same force on the other for the same amount of time. Therefore, the total change in momentum must be zero—in other words, momentum is conserved. This is why you must understand momentum.
Although the golf ball and basketball experience the same change in momentum, the smaller mass of the golf ball means it experiences a much greater change in velocity. But still, momentum is conserved. Suppose a golf ball moves towards a stationary basketball and bounces off in the opposite direction. By calculating the change in momentum of the golf ball, I can determine the recoil speed of the basketball because momentum must be conserved.
Deflecting a Bullet
Now to Wonder Woman. I want to look at the biggest possible recoil that Wonder Woman could have from deflecting a bullet. Just so that I don’t have to worry about friction, let’s suppose she is standing on ice with some bad guys that want to destroy her. A German soldier shoots at her; the bullet hits her Bracelet of Submission and ricochets at the same speed but in the opposite direction. Calculating the change in momentum for the bullet also provides the change in momentum for Wonder Woman on ice (That sounds like an ice skating show.).
To do this, I must determine the mass and speed of a bullet. After a bit of research, I found the values on this site. This puts the mass of the bullet at 4.2 grams with a speed of 965 m/s. With that, I can find the change in momentum (in one dimension):
Here I have the bullet initially moving in the negative direction with a negative velocity. When I find the difference, the negative of a negative is a positive—and the change in momentum is twice the initial momentum. Inserting the values for the mass and speed, I get a momentum change of 8.1 kg*m/s (yes, that is the unit for momentum).
Now, if the bullet experiences a change in momentum, Wonder Woman must experience the same change in momentum (but in the opposite direction). Assuming she starts from rest, her final momentum would be 8.1 kg*m/s (in the negative direction). Using a mass of 50 kg (just a guess), Wonder Woman would have a recoil velocity of 0.162 m/s (0.36 mph). That’s super slow, and that’s if she is on ice, which she isn’t. So, a single bullet striking a Bracelet of Submission would do very little to push Wonder Woman backward and nothing would happen if she were on normal ground.
Deflecting a Machine Gun
OK, so one bullet fired from a rifle might not be a big deal, but what about a barrage of bullets from a machine gun? For simplicity, I will assume this German submachine gun fires the same bullets at the same speed as the rifle, just at a higher rate. I think an estimate of 500 rounds per minute, or 1 round every 0.12 seconds, is a good figure for a World War I-era weapon. Knowing the interval between bullets, I can do more than simply find the recoil speed of Wonder Woman. I can determine the average force the bullets exert on her as they hit her Bracelets of Submission. To do this, I will use the momentum principle in one dimension. This states that the net force on an object is equal to the rate that momentum changes:
I already have an expression for the change in momentum of a bullet from the rifle example. If I use the time interval of 0.12 seconds, I get an average force of 67.5 newtons. This is about what you’d experience with a small dog sitting on you. No big deal. With just a little bit of friction, Wonder Woman could easily withstand the impact of several submachine guns. Unless she was standing on ice.
So, what kind of reaction time would Wonder Woman need to block a bullet with her Bracelets of Submission? How about a quick estimation. OK, here’s the situation: some bad guy fires at Wonder Woman with a bullet moving 700 m/s. Yes, that is a different value than I used earlier, but I want to go with a something that she should be able to block.
Wonder Woman’s reaction time depends not only on the speed of the bullet, but the distance it travels before hitting her. Instead of using the distance between Wonder Woman and the bad guy, I will use the distance at which she first notices the bullet. If you want some homework, you could use the angular resolution of the human eye and the size of the bullet to approximate the bullet detection distance, but I will call it 30 meters and be done with it.
If the bullet travels at a constant speed, I can use the definition of average velocity to find the travel time:
Inserting my values for speed and distance, I get a bullet travel time of 0.043 seconds (43 milliseconds). Not much time to react. It seems to me that a good visual reaction time might be around 160 ms—but I guess this is one reason Wonder Woman is so super.
But I can also estimate the power (in watts) that Wonder Woman would need to block a bullet. Now, blocking a bullet would require her to accelerate her hand to a given speed, then stop it and deflect the bullet—all in 43 milliseconds. It’s easier to break this into two equal parts. Part 1 is increasing the speed of the hand from zero to some value in total distance and half the total time. Part 2 is slowing the hand down in half the time and distance. Yes, this is an approximation, but it allows me to easily calculate the average velocity. Since the initial velocity is zero, the final velocity would be twice the average.
I will use a total hand moving distance of 1 meter and a hand-wrist-forearm mass of 1.5 kilograms. During the acceleration part of the hand-block, the hand would achieve an average velocity of 23.3 m/s and a maximum speed (in the middle of the motion) of 46.6 m/s. I can calculate the power with the following definition:
I already know the time interval, but what about the change in energy? Since the hand goes from moving to not moving, the change in energy is simply the change in kinetic energy:
This gives me everything I need (mass, maximum speed, time) to calculate the power. Plugging in these values, I get a bullet-blocking power of 76,000 watts. Whoa. That’s about 100 horsepower. Wonder Woman has the power of an economy car in one arm. Another reason she is so super.
All of which is to say, Wonder Woman is superhuman and you definitely should not try blocking bullets, even with the Bracelets of Submission.