Confused About Range of Football in Projectile Motion: What Am I Missing?

[Ownaginatious]

I recently encountered the following question in my homework:

"A quarterback throws a football with the same initial speed at an angle of 60 degrees from the horizontal and then at an angle of 30 degrees from the horizontal. Show that the range of the football is the same in each case. Generalize this result for any release angle 0 < theta < 2pi."

Now am I missing something here? I don't believe those would have the same range (which is the whole point of why we care about the angle in projectile motion).

Perhaps I'm interpreting the question wrong. Could someone please confirm to me that this question makes no sense, or at least explain where my thinking has gone wrong?

Thanks!

EDIT:

Nevermind; I forgot about how range decreases after you surpass 45 degrees. I feel stupid now.

 

[anathema]

Hello,

Thank you for bringing this question to our attention. I can confirm that the question does make sense and that the range of the football will be the same in each case.

To understand why this is the case, let's first define what we mean by "range" in projectile motion. Range refers to the horizontal distance traveled by an object in projectile motion before it hits the ground.

In this scenario, the quarterback is throwing the football with the same initial speed in each case. This means that the initial velocity of the football is the same in both cases. The only difference is the angle at which the football is thrown.

When the football is thrown at an angle of 60 degrees, it will have a greater vertical component of velocity compared to when it is thrown at an angle of 30 degrees. However, the horizontal component of velocity will be the same in both cases.

Now, as you correctly pointed out, the range decreases after you surpass 45 degrees. This is because at angles greater than 45 degrees, the vertical component of velocity becomes greater than the horizontal component, causing the ball to fall to the ground sooner.

But in this case, the angles given are 60 degrees and 30 degrees, both of which are less than 45 degrees. This means that the horizontal component of velocity will be greater than the vertical component in both cases, resulting in the same range.

To generalize this result for any release angle, we can use the following equation:

Range = (initial velocity)^2 * sin(2theta) / g

Where theta is the release angle and g is the acceleration due to gravity. As you can see, the range depends on the sine of twice the release angle. This means that for any release angle between 0 and 2pi, the range will be the same. It is only when the angle exceeds 45 degrees that the range will start to decrease.

I hope this explanation helps clarify the concept. Feel free to ask any further questions if you have them. Good luck with your homework!

 

[kerimek]

Hello,

Thank you for bringing up this question. It is completely understandable to be confused about the range of a football in projectile motion. The key factor that you are missing is the effect of air resistance on the trajectory of the football.

In a perfect scenario with no air resistance, the range of the football would indeed be the same for both angles of 60 degrees and 30 degrees. This is because the range of a projectile is determined by its initial velocity and the angle of release. In this case, the initial velocity is the same for both angles, so the range would also be the same.

However, in real-life scenarios, air resistance plays a significant role in the trajectory of a football. As the angle of release increases, the air resistance also increases, causing the football to lose speed and therefore, reducing its range. This is why the range decreases after surpassing 45 degrees, as you mentioned.

To answer your question, the question does make sense in a theoretical and ideal scenario. However, in real-life scenarios, the range would not be the same for both angles due to the effect of air resistance.

I hope this explanation helps clarify your confusion. Keep in mind that in science, we often simplify scenarios to understand fundamental concepts, but in reality, there are always other factors at play.