TBR Physics: "Canon Tests" Passage

[ilovemcat]

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A canon is placed 100 meters above ground level. Two balls are launched from the canon at different angles. The question asks me to compare the range distances for each canon ball launched (ie. which would travel further).

Ball A - Larger Launch angle
Ball B - Smaller Launch angle
Same launch velocity for both.

I know that when the surface is entirely flat, the maximum range is attained with a 45 degree launch angle. For this problem however, things are a bit different. Both balls are falling down... so there's a height difference to consider.

The equation for range: (Vx)(t)

With this in mind:
A larger launch angle results in more air time, but a smaller horizontal velocity.
A smaller launch angle results in less air time, but a larger horizontal velocity.

Of the two, how could you tell which change would would result in the greater range?
I'm guessing this answer is based on common sense, lol. Ball B travels further in case you're wondering.

 

[Trisphorin]

does it also depends on what exactly is a large angle and small angle? because on a flat surface, a cannon launched at 45 goes goes further than a cannon launched at 30, but if the angle were to switched to 89 and 45, respectively, the second one would obviously goes further but the first one have the larger angle. 😕

 

[ilovemcat]

does it also depends on what exactly is a large angle and small angle? because on a flat surface, a cannon launched at 45 goes goes further than a cannon launched at 30, but if the angle were to switched to 89 and 45, respectively, the second one would obviously goes further but the first one have the larger angle. 😕

For a symmetric surface, max range = 45 degrees.
Anything below or above that is lower.

For this question though, it turns out that the smaller the launch angle (ie. as angle approaches 0 degrees), the further the ball will travel. There's probably a mathematical approach you could take to prove this, but it seems far more simpler to just memorize that fact.

 

[loveoforganic]

I don't think you can common sense it - I think you have to use math. Height change is on a continuum - if the cannon is only on a 1m high platform, the ideal angle is going to be just slightly less than 45 degrees. That angle is going to decrease toward zero as the height increases.

 

[xlcooplx]

Here is the way that I understand it. Feel free to correct me if I'm wrong. The elephant in the room here is that the cannon is sitting on a cliff. Thus, even though the cannon shoots the ball out at a smaller angle, it has a larger horizontal velocity than it does vertical. Just think of a 60* and a 30* angle. They are both equidistance from a 45* angle (which is the angle at which max range is acheived on a flat surface). The 60* angle has more vertical velocity than it does horizontal velocity. The 30* angle is the exact opposite. It has more horizontal velocity than it does vertical. Since they are both the same distance from 45* (max range), their ranges will be equal WHEN the object is launched from the same height that it will land at. However, when the object's height (initial) is larger than its height (final), the object with the largest horizontal velocity will have the longest range (this will be the smallest angle). Since the 30* angle has more horizontal velocity than does the 60* or 45* angles, it will have the largest range.

I hope my crude MSPaint drawing will help explain this a little better. I know, its not exactly to scale but it will work. The dotted white line is just to show where they would land if it were launched from a flat surface.

 

[ilovemcat]

The first part of what you said sounds about right 🙂. Coop, your picture really helped. Thanks for taking the time to draw that. The dotted line was especially helpful though lol. For some reason, being able to see where they'd land on an even surface and comparing that to how they'd continue to fall when the heights are different really helps make sense of things. Thanks for the explanations guys. 😀

 

[ridethecliche]

Well done coop!

 

[xlcooplx]

Glad it could help someone. I think I remember either reading a similar problem to this or thinking up the scenario when i was going through some kinematics stuff about a month ago.

 

[BerkReviewTeach]

The first part of what you said sounds about right 🙂. Coop, your picture really helped. Thanks for taking the time to draw that.

Coop's drawing was very helpful, proving that pictures are in fact as good as 1000 words. But why didn't you just look at the answer explanation on page 40 that shows the same drawing?

Also, the problem you have listed (as you wrote it) cannot be answered. There's a reason the questions with that passage referenced specific launch angles of 0, 30, and 60 degrees. That allows us to apply the complementary angle relationship if they landed at the same elevation as they had been launched, and then add the correction associated with the extra drop distance.