Welcome to The STEM Sessions Podcast. I am Jarl Cody, your host and narrator.

We’re all familiar with scenes in movies and television shows in which a character is shot with a pistol and stumbles backwards, or is shot with a shotgun and flies backwards. It’s been overused to the point that when it doesn’t happen, the scene feels off. Well, the other evening, I saw a new spin on this troupe.

Here, the bad guy needed the good guy to cross a boundary he was unwilling to cross. So the bad guy shot the good guy once, causing the good guy to stumble across the line. He was shot in the back, so the stumble was not out of fear or reaction, it was solely from the force of the impact of the bullet.

My first thought was, “well, I haven’t seen that one before so points for creativity.” Then I realized the bad guy could have simply run ten feet to the good guy and pushed him in the back, forcing him over the line. That needless complication really annoyed me. And that annoyance will be voiced here.

This is The STEM Sessions Podcast – Episode 12. Conservation of Momentum

Imagine there is an approximately adult human-sized object a short distance in front of you, weighing 200 lb (90 kg). Now imagine you’re holding an approximately baseball-sized object, weighing about a half pound (0.22 kg). Yes, you could imagine an actual human and an actual baseball, but the precise dimensions and masses don’t matter for the sake of this discussion. We’re only concerned about the relative differences.

Your goal is to throw the baseball-sized object at the human-sized object with enough velocity to cause the latter to move away from you.

For sake of simplicity, let’s say no energy is lost to the environment via friction, air resistance, heat generation, etc. Therefore the velocity of the baseball-sized object as it leaves your hand is the velocity with which it hits the human-sized object, and no fricition needs to be overcome for the human sized object to move. The object sitting on ice is the classic example.

Further, we’ll assume the two objects stick together after the collision and no mass is lost. In other words, the baseball-sized object does not break apart, the human-sized object doesn’t chip, and no energy is lost as heat during the collision.

Finally, we’ll assume the impact point is the human-sized object’s center of mass. Thus, we don’t have to worry about angular momentum, and all motion is linear to the initial vector of the thrown object.

So how fast do you need to throw the baseball-sized object?

Before we dive into the math, let’s work through it conceptually. The human-sized object is roughly 400 times heavier than the baseball-sized object. This is a two orders of magnitude difference, because we know from a previous episode, an order of magnitude is power of 10, and 400 is two powers of ten.

Therefore, intuitively we know it will take a significantly higher force to make the human-sized object move at a given speed than it would the baseball sized object. We know this by the relationship between mass and force in Newton’s Second Law of Motion, F=ma. As mass increases, the force must increase to produce the same acceleration. In our case, a two orders of magnitude increase would be a good starting point.

So let’s say you throw the ball with a speed of 60 mph (97 km/hr). If we use the two orders of magnitude as an indicator, the merged entity will travel at approximately 0.6 mph (1.0 km/hr). For reference the average walking speed of a human is 3 mph (4.8 km/hr). That’s a fifth the speed of someone walking.

Is 0.6 mph the actual result? Of course not. It’s just an approximation based on differences between the starting values. But it confirms the difference in masses of the objects has a tremendous impact on their behavior after a collision, and it’s a good rule of thumb for predicting that behavior. For the exact predictions, we turn to the conservation of momentum.

This principle states the total momentum of a system before an event will equal the total momentum after the event. In our case, the event is the collision between the two objects. Therefore, the sum of the momentums of the objects before the collision must equal the momentum of the merged entity after the collision.

Conservation of momentum is derived from Newton’s Third Law of Motion, commonly refered to as “equal but opposite forces”. More accurately, it can be read as the sum of the forces in a static system must equal zero, and in the case of our collision, if no energy is lost or added, then the sum of the forces before and after the collision must be equal.

From forces, we can derive impulses (force times the duration of the event). And because the duration is the same for both objects, the impulses must be equal. From impulses, we can derive momentum (mass times velocity), which also must be equal.

In the moment before the collision, the baseball-sized object has a momentum of 0.2 kg (its mass) times 27 m/s (its velocity), which equals 5.4 kg m/s, and the human-sized object has a momentum of 90 kg (its mass) times 0 m/s (its velocity), which equals 0 kg m/s. Therefore, the total momentum prior to the collision is 5.4 kg m/s plus 0 kg m/s or 5.4 kg m/s.

In the moment after the collision, the merged entity (remember, we’re assuming the objects stick together) will have a mass of 0.2 kg (mass of the baseball-sized object) plus 90.7 kg (the mass of the human-sized object) or 90.9 kg. To find the resulting velocity, we divide the initial momentum of the system (5.4 kg m/s) by the final mass (90.9 kg), and determine the entity will be traveling 0.06 m/s (0.22 km/hr or 0.13 mph).

Again, the average walking speed of a human is 3 mph (4.8 km/hr or 1.3 m/s), so the merged entity is traveling at about 5% of a human’s walking pace. This is about 1/5 the speed we predicted in our earlier estimate.

From here, it’s easy to run off other scenarios. If you want the final velocity to match a human’s walking pace (effectively increasing the velocity 20 times), then the initial velocity of the baseball-sized object needs to increase by a similar amount. You could also increase its mass and accomplish the same result, because the momentum increases proportionally with increases to either the mass or velocity. It becomes a bit more complicated if the human-sized object also has an initial velocity, but the relationships between the terms stand.

The proportional scaling works in the other direction, too. Instead of a baseball-sized object, let’s say you have a pebble with a mass of 15 g (0.015 kg or 0.03 lb). This pepple is only 6.6% the mass of the baseball-sized object. Assuming it is also thrown at 60 mph (97 km/hr or 27 m/s), its initial momentum will be 6.6% that of the rock or 0.4 kg m/s. And completing the estimates, the resulting merged entity’s velocity will also be 6.6% that of the first scenario for a final velocity of 0.004 m/s or 0.014 km/hr or 0.009 mph. That’s effectively zero velocity, all because the mass of the pebble is completely dwarfed by the human-sized object; 0.01% to be more precise.

I selected a pebble mass of 15 g for a reason. It’s the mass of a .45 caliber bullet, which will show us the absurdity of the Hollywood trope of someone being knocked backwards when shot.

The velocity of bullet depends, on the amount of powder in the cartridge among other factors, but a bullet fired from a .45 caliber pistol will top out at 1000 ft/sec (680 mph or 1100 km/hr or 300 m/s). This is 11x faster than the 60 mph objects in our previous examples. Surely that will have significant impact…

The momentum of the bullet is 4.5 kg m/s, yet the resulting speed of the bullet-human entity is only 0.05 m/s (0.18 km/hr or 0.11 mph). That’s still a fraction of a normal walking pace – about 3.5% to put a number to it.

For the sake of completion, let’s estimate the velocity of the Hollywood jerk-back effect to be 2 m/s (7.2 km/hr or 4.5 mph). This is likely an under estimate for some scenes. Using that velocity, the final momentum of the bullet-human entity is 2 m/s times 90.715 kg, or 181 kg m/s. Prior to being shot, that human’s momentum was zero (no velocity), so only the bullet had momentum. It’s initial velocity is therefore the final momentum (180 kg m/s) divided the mass of the bullet (0.015 kg) for a velocity of 12,067 m/s (39,590 ft/s or 43,441 km/hr or 26,996 mph)! That’s 35 times the speed of sound. Mach 35‼ For reference, the fastest commercially loaded bullet, a .17 Remington, travels at 4,400 ft/s which is only 1340 m/s. That’s only 11% of the velocity required to get knocked back by a bullet.

The point of this exercise was not to bust a long-standing action movie troupe, but instead to show you can make sense of a lot of what you see in reality, or on the screen, with only a basic understanding of physics and scaling factors. From the equations F=ma and p=mv, the relationships between force, momentum, mass, and velocity are seen. If a variable on side of the equation increases, so must the other side. The heavier an object, the greater the force required to move it. The heavier an object, the more momentum it will carry. Knowing the quantities for each variable isn’t necessary. It’s the relationships that matter more.

Thank you for listening to The STEM Sessions Podcast.

This episode was researched, written, and produced by me, Jarl Cody.

While I strive for completeness and accuracy, I encourage you to do your own research on the topic we discussed, and confirm what I’ve presented. Corrections and additional information are always welcome.

Shownotes, contact information, and details of our other activities can be found on our website thestemsessions.com

If you received value from this episode, and wish to give some back, please visit thestemsessions.com/valueforvalue for ways to support the podcast.

Finally, please remember STEM is not the exclusive tool of experts, policy makers, or talking heads. Every presenter is susceptible to unconscious and, sometimes, deliberate bias, so always verify what you read and what you’re told.

Until the next one, stay curious.

REFERENCES

Wikipedia.com, “Speed of Sound”, https://en.wikipedia.org/wiki/Speed_of_sound

WideOpenSpaces.com, “8 Cartridges with the Highest FPS Speed”, https://www.wideopenspaces.com/8-cartridges-highest-fps-speed-pics/

PhysicsClassroom.com, “The Logic Behind Momentum Conservation”, https://www.physicsclassroom.com/class/momentum/Lesson-2/Momentum-Conservation-Principle