Momentum and Football

I think we are entering a new era. An era where it is quite simple to find and get great videos. Oh, just saw a great tackle on the TV? In the old days, you would have to get that video off the TV yourself. Not anymore. Welcome to the interwebs. Also, the quality is awesome compared to 10 years ago. Here is the video. Yes, I know this is from several weeks ago - I am slow. Also, thanks to the person that put this on youtube - I edited your clip to remove the music and just look at the one collision. You did a good job though.

In terms of video analysis, this isn't too bad of a video. Yes, there is a problem - if I call the direction of motion the x-axis and the direction parallel to the yard lines the y-axis then these do not look perpendicular. Oh well, the motion is mostly in the x-direction so I will proceed. Tracker Video Analysis is awesome for this video because of the "calibration point pair" feature. Essentially, the calibration point pairs allow me to mark some locations on the field in each frame. This will adjust the origin, the scaling and the angle of the video. You can make more than one pair so that when your original pair goes out of the frame you can continue. (Note: I found out that Tracker Video does not work with QuickTime X - this is due to Apple, not Doug - the creator of Tracker.)

I proceeded with the video analysis anyway. At least I had something to scale the video easily (the yard markers). Here is the down the field motion of the player (I marked his waist in each frame).

I fit a linear function to the position data for the time before the collision. This gives a fairly (I was surprised) constant speed of 6.2 m/s (about 14 mph). Here is the fit to the data after contact.

The speed of the two players stuck together is 3 m/s. I still can't believe that everything turned out this nice. Well, what can I find from this info? Suppose I make some assumptions. Assume that the collision lasts a very short time. Assume that the initial velocity of the second football player is 0 m/s. Assume that the two have the same final velocity (I didn't actually measure that). Here is a diagram of what is happening.

The important idea here is the momentum principle.

If I look at the change in momentum from the time just before the collision to just after, then this is a good diagram to consider: (I will label the white object as "a" and the dark red as "v")

I could apply the momentum principle to both the "a" and the "v" objects. However, there is something they share. (also note my force convention. F_a-v is the force on v from a). The force of a on v is the same magnitude (but opposite direction) as v on a. This is just because they are the same interaction. Also, the time that v pushes on a is the same time that a pushes on v. Finally, I put "other forces" on both of the objects. These could be small (and likely are), but I will leave them there. Here is the momentum principle for the two objects.

Now, I will put in the second equation negative the force on a for the force on b:

Adding the two expressions, the last terms cancel:

Now, what about these "other" forces. First, these are really "other-net forces". If these were objects sliding on a frictionless surface, the net forces would be zero (vector). But what if they (football players) are both pushing on the ground? Well, then they would be pushing in opposite directions and partly cancel. Finally, the key is the time interval. No matter how long or short the time interval is, the the a-v, v-a forces times delta t cancel (because they are really the same). If the time interval is very short (typical for a collision), then this other stuff just doesn't contribute much. I am going to let it be approximately the zero vector. This leaves me with:

Ok, so now for this particular situation. Let me just look at components in the x-direction. If the initial momentum of v is zero and the two players have the same final velocity, then:

You can see I solved for the mass of player v in terms of the mass of player a. This does have the correct units, and it gives a positive mass (because he was moving faster before he hit the v-guy). Since I have some numbers for the velocities, I get that the v-player has a mass of:

See. Physics works (assuming the v-player's mass is a little bit larger - I don't really know for sure).