Yup. He just did that. He kicked the football from the opponent's 40 yard line. That is 60 yards to the endzone plus 10 more yards to the field goal post. Obviously, this kind of kick isn't too easy. If it was, you would see things like this in college and NFL games. Ok, let me correct that. After looking at the Wikipedia page on field goals, it appears that there have been some field goals made from over 60 yards (there was one from 69 yards). Well, it's still difficult.
Let's say you can kick a 40 yard field goal. How much more difficult would it be to kick a 70 yard field goal? There are really two things you need to do when kicking the ball. You need to give it an initial speed and you need to aim it. You have to aim by adjusting the "left-right" angle as well as the "up-down" launch angle. For now, let me ignore the left-right aiming.
Why is this a difficult physics problem? It's difficult because there is more than just one constant force on the ball while it is in the air. In your introductory physics class, you looked at projectile motion. In a projectile motion problem, you assume that the air resistance is negligible. This means there is only the constant downward force of gravity. The ball will then have a constant x-velocity and a constant acceleration in the y-direction. It might seem complicated, but it really isn't so bad.
For a football, we can't ignore the effects of air resistance. Here is a diagram from the last time I talked about football trajectories.
In this model, the air resistance force is proportional to the square of the ball's speed (with respect to the air). This means that the forces are not constant and you can't use the standard kinematic equations commonly used in projectile motion problems. Actually, I am pretty sure there would be at least one other force on the ball - some type of lift force. I really don't have a good way to model this (yet), so I will leave it off. Oh, someone has some nice video data showing the motion of a football - but I don't have it yet.
Even though this is missing a force, I think it will still give an idea of how fast you have to kick the ball and at what angle in order to get a field goal. Here is how we can deal with these non-constant forces - with a numerical model. In a numerical model (or numerical calculation), the motion is broken into many small time steps. During each step, I can make the assumption that the air resistance force is constant. This isn't such a terrible assumption if the time step is small. I know it seems like it is cheating, but the simple truth is that if it works, it works.
Here is how to do a numerical calculation for a kicked football.
- Based on the current velocity, calculate the air resistance plus gravitational force.
- Use this force to find the change in momentum over the small time step due to this net force (and the new momentum).
- Use the momentum (and thus velocity) to find the new position of the ball.
- Update the time and repeat until the ball gets to the ground.
It's really that simple. I will leave out the rest of the details - you can see this post if you want to learn more.
Once I can model the motion of a football (which I still can't do since I don't know the lift force), I can play with this stuff. First, what is the best angle to kick the football at? If I am kicking the ball really slow, a 45° would be close to the best angle. At an angle of 45°, you get the best of both horizontal velocity and time for the ball to be in the air. Here is a more detailed derivation of the maximum range with no air resistance.
For the football, what is the best angle? Here is a plot that shows the best launch angle for different launch speeds.
This looks jagged because I only changed the launch angle in 2° increments (with a smaller angle increment, this would take much longer). Here you can see that with a low launch speed of 25 m/s, you would kick much closer to 45°. Over 55 m/s, that angle would be down to around 36°. Of course, there are other factors you would have to consider in an actual football game. If you kick the ball too low, there is a greater change the kick can be blocked.
Now what about the launch speed? Based on this best angle, here is a plot of the speed needed for different kick distances. And yes, I did take into account the fact that the goal post is 10 feet above the ground. This shows the velocity needed to get right to this goal post (ignoring wind). Obviously, you would want to kick it a little bit faster to make sure you clear this post.
Here is a quick preemptive comment complaint. Yes. I have the velocity in m/s and the distance in yards. That might seem odd. However, look at the football field. What are the distances measured in? Yes, yards. It really only makes sense to plot the distance in yards. I will not plot the speed in yards per second because that's just dumb.
I guess this plot really just says the obvious. If you want to kick the ball farther, you have to kick it faster. How about something less obvious? How hard would you have to push on the ball during the kick? Let's make some crazy assumptions about a football kick - starting with this diagram.
Let's just say that the change in height of the ball during the kick is small enough to ignore. Further, I will estimate that the foot pushes on the ball a distance of 1.5 meters. If I think about the average force from the foot on the ball, I can say that this force does work on the ball to change its kinetic energy. The work-energy principle would then say:
Based on the mass of the ball, here is a plot of average force as a function of launch speed.
If you want to double the launch speed, you would have to quadruple the average kicking force. I guess that plot isn't as useful as a plot of the average kicking force as a function of field goal distance. I mean, who cares how fast the ball is going? We only care if it passes through the uprights.
From this, I get an average force of 96 Newtons to kick a 40 yard field goal and an average force of 247 Newtons for a 70 yard field goal. Oh, you want to kick a 100 yard field goal? That would take an average force of 544 Newtons. Boom. That's the sound of the ball exploding. I'm just kidding. I have no idea how strong these balls are.
