Vectors and Projectiles Legacy Problem #9 Guided Solution
Problem*
Avery, the quarterback of South's Varsity football team, made the most amazing pass in the Homecoming game against cross town rival North. He threw the pass from the exact center of the field to the corner of the end zone, where Jamaal caught it for the game winning score. If the football field is 160 feet wide (sideline to sideline) and it is 60 yards from midfield to the back of the end zone, then how far did the ball travel from Avery's hands to Jamaal's
Audio Guided Solution
This problem is going to require some diagramming, some thinking, and some careful attention to units. We'll discuss the units first. There are two dimensions given to this question. One of them is in feet and the other one is in yards. And feet and yards just don't mix, so we need to get them in the same units. The best way to do it, perhaps, is to take the 160 feet and divide it by 3. And you'd end up with 53.3 repeating yards as that other dimension. Now that's the dimension from sideline to sideline. So when it comes to the diagramming part of this problem, I'm going to draw a picture of a football field. It's kind of a long rectangle, narrower than it is long. I have a football that's being thrown from the exact middle of that football field, middle, horizontally and vertically, to the very corner of the end zone, which is one of the corners of our rectangle. So I draw a line across the middle of the football field. That's the 50-yard line, as it's sometimes called. And the ball is thrown from the middle of that middle line to the exact corner of the football field. So if you were to take the 160 feet and divide it by 2.5, you'd have the middle of that football field being 80 feet from a sideline. And if you take that 80 feet and divide it by 3, you get 26.6 repeating yards. The ball has a displacement that goes horizontally 26.6 repeating yards, and vertically 60 yards to the very corner of the end zone. Now I would draw that little right triangle that has those two dimensions to it, and I'd draw the hypotenuse of the right triangle that goes from the exact middle of the field to the corner of the football field. What I'm looking to find is the hypotenuse of that right triangle that has as its two sides 26.6 repeating and 60 yards. Now that involves the use of the Pythagorean theorem. So I use my Pythagorean theorem, a squared plus b squared equals c squared, to find the length of the hypotenuse. And that's all there is to this problem.
Solution
66 yd (rounded from 65.66 yd) or 197 ft
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities in an organized manner. Equate given values to the symbols used to represent the corresponding quantity - e.g., \(v_\text{ox} = \units{12.4}{\unitfrac{m}{s}}\), \(v_\text{oy} = \units{0.0}{\unitfrac{m}{s}}\), \(d_x = \units{32.7}{m}\), \(d_y = \colorbox{gray}{Unknown}\).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
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