There are 20 seconds left on the clock. Your team is down by 2 points such that a field goal would win it. The ball is spotted on the hash mark at the 15 yard line and it is first down. What to do? Should you call a run play so that the ball is in the center of the field? Or should the ball be kicked from where it is?
So there is the question. Is it better to kick the ball from an angle or move back and kick it head on? Let me just look at one aspect of this situation. What is the angular size of the goal post from the location of the kicker? I am not looking at the height of the horizontal goal post - I will assume the kicker can get the ball over this.
Let me start with a simple diagram. There are really three points of interest. The location of the two posts (and by location I mean the x-y coordinates) and the location of the kicker (or the ball).
Here there are three important points. The location of the two goal posts (as seen from above) and the location of the kicker (represented by my attempt at a football). You can also see the angle θ that shows the angular size of the goal as seen from the kicker. The two vectors r1 and r2 are needed to find the angle theta. How will I do this? Dot product (or scalar product). Ok, what is the dot product? Let me just say that it is one way to operate two vectors. You could think of it the product of the magnitude of one vectors and the component of the other vector. Here is a diagram of the dot product for two vectors.
This would give a value for the dot product as:
Here is the key. I can use this dot product to find the angle between these two vectors. How? Well, I can use the other definition of the dot product that is the sum of the product of the components of the two vectors. You can see that this is the same as above if one of the vectors has only one non-zero component. For those same two vectors, I can write:
Quick note: with this definition you can see that the order of operations does not matter (this is not true for the cross product).
Ok, here is the plan:
- Get the three points (two for the goal posts and one for the kicker).
- Determine vector r1 from the points of one of the goal posts and the kicker.
- Do the same for vector r2.
- Calculate the dot product using the components of these two vectors (the second dot product definition).
- Determine the magnitude of vector r1 and r2.
- Use the first definition of the dot product to calculate the angle θ
- Do the above for a whole bunch of different points on the field and see how the angular width of the field changes.
What about the details? What are the locations of the goal posts? Where can the kick be placed? According to Wikipedia, the goal posts are 18 feet and 6 inches apart. Hash marks are also important. These are the boundaries of where the ball can start. In NCAA college football, the hash marks are 40 feet apart. In the NFL, they are just 18 feet and 6 inches wide. Really, that is all I need. Now on to the plot (which took way longer to make than I would have expected).
That does look pretty, but what does it mean? Well, the z-direction represents the angular size of the goal posts. The goal posts are on the y-axis and the x-direction is to the right. So, the data here starts at x = 30 feet - that would be the goal line.
Where is the maximum angular size? Of course it is at the goal line in the center of the field. And yes, this is just the horizontal angular size. This does not include the horizontal cross bar (remember, I stated that at the beginning of this post).
Ok, back to football. Suppose the ball were placed on the left hash mark at the 10 yard line. Would you be better off downing the ball in the center of the field with a 3 yard loss? So, which has a better angular size for the goal post? One thing I didn't include in the above calculation was the location of the ball for the kick (duh). Typically, the ball is snapped about 6 yards behind the line of scrimmage. This means you wouldn't ever be kicking the ball from the goal line. The closest kick would be at about the 6 yard line. So, if the ball starts at the 10 yard line at the hash mark and kicked from the 16 yard line, the angular size of the goal would be 0.222 radians. If the kick was moved back three yards and to the center, the angular size would be 0.212 radians. So, not a good move.
Is it ever better to move back and in the center of the field? Ok, how about a graph. How about a graph of the change in angular size in moving from the hash mark to the center of the field and 3 yards back? I can do that. In fact, this is a plot for a loss of 1, 3, and 5 yards while moving to the center.
If you are going to lose just 1 yard, then the kicker will essentially always create an increase in angular size by the move. However, if your lateral move is going to cost 3 or 5 yards the angular size of the goal will decrease. In other words, the increase in angular size due to better angle will be less than the loss of size due to being farther away.
Notes: Some quick points to mention.
- In the NFL, the hash marks are the same width as the field goal posts. I can't imagine (nor do I remember seeing) this lateral pre-field goal kick move in the NFL.
- There might be some other advantages to moving. Maybe your kicker hooks the ball to the left (or right). This might make repositioning a benefit. Although the kicker is shooting from an angle, the offensive line is still parallel to the goal posts. That might be important.
Preemptive comments
Listen. I know people have trouble with our current commenting system, so I want to help. Let me go ahead and post some comments for you.
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- Dude. You know nothing about football. Teams make that move just to burn time off the clock. Play a sport and stop being a dork.
- Ummm....I think you are confusing Futbol with American Football. There can be only one.
- You failed to take into account the ball spin and the rotation of the Earth (you know, Coriolis Effect)
- What if someone blocks the kick? Wouldn't that make the kicker miss?
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