I was working with a physics student the other day and came across a football trajectory problem which was very timely because Superbowl Sunday fast approaching and the Pittsburgh Steelers are still in the mix. Yeah!

Most trajectory problems in physics problems meant to be worked “by hand” have to make some assumptions so that the equations can be solved without using a computer. The biggest and most common assumption is that the effect of the air resistance on the object is negligible. “Negligible” means that we consider (or at least are willing to temporarily pretend) that the effect is small enough that if we neglect accounting for it in our calculations our answer could still be useful in some way. For example, if:

a) we know our kicker can usually kick with an certain initial speed v, and if

b) we know how to use basic trajectory physics calculations to tell us the maximum possible range R for a kick at a given angle from the horizontal (let’s call it theta), and if

c) we know how to use calculus to optimize our choice of initial conditions for various purposes, such as maximizing range, or height above the cross bar, and if

d) we are willing to accept an answer whose mileage may vary depending on how the ball is kicked (a tight spiral is best, but end over end is next best) due to the variety of magnitudes of air resistance, and if

e) we are willing to recognize that our theoretically calculated range is in practice greatly reduced due to air resistance forces (drag) by an additive quantity which is some factor times the square of the result of taking the ball’s forward speed minus the wind’s forward speed, with both speeds being measured with respect to the ground, and

f) point e) applies to a greater extent as for initial velocity and final range increase because drag forces and the resulting decelerations factor in more heavily as the speed and its square increase (therefore drag typically especially decelerates the ball during the initial fastest moments of the trajectory)

Note that in the coordinate system I am defining for these problems, a headwind (kicking into the direction the wind is coming from) will be considered a negative wind speed while a tailwind (kicking with the wind) could be considered a positive wind speed. That way, the difference between the ball’s (also defined as positive in the forward-going direction) forward speed and the wind’s forward speed increases to a value higher than the ball’s speed relative to the ground in a headwind, and decreases to a value less than the ball’s speed relative to the ground if kicked in a tailwind.

We can also see that in a very unusual special situation where the tailwind is roughly constant, in the same direction of the kick and equal to the ball’s initial horizontal ground speed, the assumption of negligible air resistance is much more reasonable, especially for fairly low angles of kick (drag forces would still be acting in the vertical equation of motion, even in this situation). Although highly unlikely this is theoretically possible. In fact, to get an idea of how much wind would be reasonable to hypothesize, I searched the web for "windiest NFL game" and found a hilarious video of some field goal attempts gone wild during an NFL game that was played in winds gusting to 45mph, which is fairly close to a typical kickoff’s initial horizontal velocity. Such weather could happen again, and if the wind’s direction and speed happens to line up with the kicker’s targeted direction, we can expect negligible drag predictions to actually become fairly realistic. In such ideal conditions, for example one might expect a kick with an initial velocity of 30m/s to fly around 30 yards further (between 80 and 85 yards) before returning to earth than the identically launched kick would travel under calm conditions. I would guess that wind conditions were favorable when the records for longest field goal (63 yards for the NFL, 62 yards CFL) and longest punts (98 yards NFL, 108 CFL) were set. Note that with the longer field (110 yards) and the rules for recording a punt, the difference in best punt likely has more to do with how the statistics for punting are kept than the actual distance the ball traveled -- there may have also been some lucky bounces involved).

However for typical football weather, we would simply expect our answer to over-estimate the range because in realty the football simply does not have enough mass to avoid letting the small forces due to air resistance (also known as drag) apply a negative horizontal acceleration (or a slowing effect) to the ball’s trajectory, especially for long punts, field goal attempts or hail mary passes.

Let’s take a brief side trip to examine why the football’s relatively low mass means neglecting air resistance throws off our answer more than if the football were heavier. We will compare and contrast the flight of a shot put and a football with identical initial speeds and launch angles in air and in a vacuum. In a vacuum, with no air resistance one would expect both objects to travel identical trajectories, but in air, we would expect both objects, especially the football, to slow down and fall short of that ideal. Let’s apply Newton’s second law of motion, F = m*a (where F is the force acting on an object, m = the mass of that object, and a = the acceleration, or rate of change of its speed, of an object) to examine this more closely. I will assume that the drag force on the shot put and the football are about equal when the football is thrown in a tight spiral because they have roughly the same cross-sectional area when viewed from the front, and aerodynamicists say that drag forces on an object moving through a fluid are typically proportional to cross-sectional area, fluid density, and the square of v_fluid, where v_fluid represents the velocity of the object with respect to the fluid. There is also another factor called the drag coefficient that accounts for the effect on drag of an object’s shape but for the purposes of this discussion we will consider the drag coefficients for a tightly spiraling football and a shot put to be close enough not to worry about. To simplify comparing the acceleration due to air’s drag on a football with the acceleration due to air’s drag on a men’s Olympic shotput we can solve F = ma for a (by dividing both sides by m) to write the same law in a different form:

a = F/m.

This expression for acceleration shows that the resulting accelerations of the objects due to drag will increase in direct proportion to the drag forces acting on the objects. In our case we are assuming that the drag forces on the two objects are roughly equal, so this is of not much concern. The equation also shows that the resulting accelerations of the objects due to drag are inversely proportional to the masses of the objects. In other words, if the weight of a football ranges around 14-15 oz | 0.9 lb |0.4 kg, and the weight of a (men’s Olympic size) shot put is 16 lb | 7.26 kg (women’s is 8.8 lb or 4 kg), then the roughly 16 times more massive shotput will experience roughly one sixteenth as much acceleration due to drag as the football. I chose not to use the word deceleration here despite recognizing that these drag forces, and therefore the accelerations due to drag, always oppose the motion of the objects relative to the fluid, because in the case of a tail wind, “drag” forces actually impart a positive acceleration to the object; more on that later. The main point of all this is that we can conclude that neglecting drag always gives erroneous results when calculating trajectories of object’s moving through fluids, but the magnitude of the error is inversely proportional to the object’s mass. We would expect drag to shorten a football’s flight trajectory somewhere between ten and twenty times more than a men’s Olympic shot put thrown with the same initial velocity.

Having given a little justification for the common practice of assuming negligible air resistance, in my next blog I will discuss how to determine the general formula for the expected range when air resistance is neglected. I'm planning to follow that up with another blog outlining the procedure for determining the optimal launch angle is (assuming negligible air resistance). Later, if anyone expresses interest, I’m prepared to discuss the general procedure necessary to take wind and drag into account; practical basic computing techniques will be involved in developing a simple physics simulation. If there is any interest I may even eventually write a companion entry touching on ordinary differential equations and their digital corollary, difference equations, and how they apply to this kind of simulation.

If you enjoyed this post and would like to see more on this or similar topics, please let me know by sending me a message or leave a comment for all to share.