The Projectile Motion of Throwing a Football

By: Chad Hainley (cmh5503) and Billy Jack (wjj5015)

Projectile Motion

• Projectile motion can be described as a vector and be broken up into two components:– X component– Y component

X Component

• The X component is the velocity of an object from side to side

• In perfect conditions, this velocity will remain constant throughout its projected motion

• This is because in perfect conditions there are no outside factors acting on it

Y Component

• The Y component is the velocity of the object as it moves up and down.

• Unlike the velocity in the X direction, in the Y direction, the velocity continually changes.

• This is because gravity is acting upon it, decelerating it at 9.8 m/s. This is what gives the object the arced projected motion instead of it just flying straight forever.

Example

• Let’s say that Matt McGloin is under center at his own 45 meter line and that there are 2 seconds to go in the 4th quarter with his Nittany Lions down by 6 to the Buckeyes. He knows that he needs to get a touchdown which is 55 meters away. He also knows that he will have to drop back in the pocket 5 more meters to give himself ample time to make a play which will take him about .5 seconds. In perfect conditions, how fast and at what angle will McGloin have to throw the ball to hit “My Boy Moye” in the endzone if he is blazing down the field at 12 m/s?

Example• First we need to find the X component. To do this we must

find out how long it will take for Derek Moye to make it to the endzone. Since we know the distance and the speed we can just divide the distance by his velocity:– 55/12 = 4.583 s

• Now that we know this we need to find out how fast in the X direction the ball must travel to reach Moye on time. BUT! Let’s not forget that McGloin will be retreating in the pocket for the initial .5 seconds so we must subtract that from the time found in the previous step. Now we just divide the distance between McGloin and the endzone by the new time:– 60/4.083 = 14.695 m/s – our X component!

Example

• Now to find the Y component! To do this we must figure out how fast in the Y direction the ball must be going in order to be in the air for the needed amount of time of 4.083 seconds. To make calculations easier we will cut the time in half so it is just until the ball reaches the peak of its trajectory. – 4.083/2 = 2.0415s

• Using this time and our knowledge of the force of gravity we can find the Y component using the formula:– acceleration = (final velocity – initial velocity)/time

• -9.8 = (0 – V)/2.0415• -20 = -v• Our Y component is 20 m/s!

Example

• Now that we have both components we can find the velocity and angle!

• To find the velocity vector all we need to do is use the Pythagorean Theorem– V2 = 14.0952 + 202

– V = 24.468 m/s

Example

• Now to find that angle!• To do this we just take the inverse sine

of the Y velocity over the hypotenuse:– Sin-1 Θ = (20/24.568)– Θ = 54.5°

Answer

• Since our QB is awesome and has much knowledge of physics he throws the ball precisely at 24.468m/s at a 54.5° resulting in a game winning touchdown!!

Why is a Football a Spheroid Though?

• What wasn’t discussed in class however, was the ease at which an object is able to pass through the air based on its shape.

Explanation

• The spheroid shape of a football allows the nose of the football to “cut through” the air and move in the arc that it does, rather than to have the air “fight” against the ball and force it down early.

Explanation• The spin of the football also allows for the football to be

more aerodynamic.• The spin of the football stabilizes the football through

angular momentum and torque, which allows it to travel in a tight spiral.

• When thrown properly, the ball gains an orientation that gives it the smallest possible cross-sectional area against the oncoming air, which causes the least amount of aerodynamic drag.

• If the ball does not spin properly (a wobbly pass), air travels excessively under its tip as it descends, causing it to tumble and lose some of its forward momentum due to a greater cross-sectional area being exposed to the wind.

So That’s why a Football is a Spheroid!

• With the aerodynamics of the football being very important to every quarterback, they can thank the shape of the football for its ease to spin around its axis and its nose to penetrate through the air easily (if thrown properly).

• With the spin of the football and nose penetrating through the air, the aerodynamics of the football are vital to any quarterback.

Conclusion

• In conclusion, understanding both the projectile motion and aerodynamics of the football can make any physics fan also a football fan! (Or vice-versa)

• Now only one question remains . . . Who will be Penn State’s quarterback next season?