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Sunday Night Trigonometry

We posed this question in a social media quiz: A 45-yard field goal attempt is flying at 25 yards/second at an angle of 40 degrees above the horizontal. Assuming minimal air resistance, how many points will the team score?
Football fans often have a love-hate relationship with their team’s kicker. For pure math fans, there’s plenty to love in every kick, regardless of whether the team scores.
Through five weeks of the 2017 NFL season, kickers had a field goal completion rate of just more than 85 percent, with the longest sailing 61 yards for Eagle’s rookie Jake Elliott.
Here’s how to answer the question we posed – assuming little impact from the wind and discounting the lift created by the ball’s spin so this doesn’t become a complex homework assignment.
Average NFL kickers can send the ball flying at 55 mph (25 yards per second) and they need to start the ball at about 40 degrees above the ground to get up and over the defense.
Because of the Earth’s gravitational pull, the ball will accelerate down in the y direction. If we ignore air resistance there is nothing pushing or pulling the ball horizontally, so it will move with a constant velocity in the x direction. We need to analyze these two types of motion separately. Luckily, we can use some trigonometry to split our velocity into x and y components.
Here’s the math:
Split the x and y dimensions:
V = 25 yard/s
Vx = 25 y/s COS (40o) = 19.15 y/s
Vy = 25 y/s SIN (40o) = 16.07 y/s
a = 10.7 y/s2 toward the ground
Next, determine the time necessary to go 45 yards – the length of our imaginary field goal attempt.
Find the time to go 45 yards:
Vx = (Delta*X)/t
So, t = (Delta*X)/Vx = 45 yards/19.15 y/s = 2.34 seconds
With all that information – height, distance, drop rate and speed – calculate how high the ball will be above the ground when it reaches 45 yards.
Find the ball height at 2.34 seconds
or 0.5(-10.7 y/s2)(2.34s)2+(16.7y/s)(2.34s)+0 = 9.78 yards
Our own math content expert Stephen Jehl did the mathematical legwork for us and found that the ball would be 9.78 yards above the ground when it reached 45 yards. The goal post cross bar is 10 feet tall – or 3.3 yards – so the kick was a success.
NMSI strongly advocates showing students how math and science impact their everyday lives. For those interested in football, the math of a field goal attempt is a fun way to capture their attention.
Want more? NMSI offers three-year, scaffolded training, as well as classroom-ready lessons and assessment tools for teachers in grades 3-12. Check out our Laying the Foundation work and Contact Us if you would like to get training to your school.


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