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Mimi_x3
Group Title
A cannon shell is projected at \(u\) m/s at elevation angle at \(\theta\).
The cannon hit the target at \(X,Y)\) when fired at two different angles \(45\) degrees and \(63°26'\). Show that the angle of elevation of the target from the gun is given by \(\beta = tan^{1} \frac{1}{3}\)
 2 years ago
 2 years ago
Mimi_x3 Group Title
A cannon shell is projected at \(u\) m/s at elevation angle at \(\theta\). The cannon hit the target at \(X,Y)\) when fired at two different angles \(45\) degrees and \(63°26'\). Show that the angle of elevation of the target from the gun is given by \(\beta = tan^{1} \frac{1}{3}\)
 2 years ago
 2 years ago

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Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
bad latex ._.
 2 years ago

Mimi_x3 Group TitleBest ResponseYou've already chosen the best response.1
something is wrong with this site; its not suppose to fail
 2 years ago

Mimi_x3 Group TitleBest ResponseYou've already chosen the best response.1
A cannon shell is projectile at \(u\) m/s at elevation angle at \(\theta\). The cannon hit the target at (\(X,Y)\) when fired at two different angles \(45\) degrees and \(63°26'\). Show that the angle of elevation of the target from the gun is given by \(\beta = tan^{1} \frac{1}{3}\)
 2 years ago

Mimi_x3 Group TitleBest ResponseYou've already chosen the best response.1
better now :D
 2 years ago

Mimi_x3 Group TitleBest ResponseYou've already chosen the best response.1
dw:1344003862684:dw
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
\[x = u \cos \theta \cdot t\] \[y = u\sin \theta \cdot t  \frac12\cdot g\cdot t^2\] \[v_y u_y = gt \implies u\sin \beta  u\sin \theta =gt \implies \frac{u\sin \beta u\sin \theta}{g}\] \[\frac{y}{x} = \tan \beta\]
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
\[t = \frac{u\sin\theta  \sin \beta}g\]
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
From the first two equations and the last one. \[\frac{y}x = \frac{u\sin \theta\cdot t\frac12gt^2}{u \cos \theta\cdot t} = \tan \theta  \frac12\cdot g\cdot\frac{t}{u\cos \theta}\] Substitute the expression for t.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
\[2\tan \beta={\tan \theta} + \frac{\sin \beta}{\cos\theta}\]
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
\[2\tan \beta = \tan \theta_1 + \frac{\sin \beta}{\cos \theta_1}\] \[2\tan \beta = \tan \theta_2 + \frac{\sin \beta}{\cos \theta_2}\] Solve it for \(\beta\).
 2 years ago

Mimi_x3 Group TitleBest ResponseYou've already chosen the best response.1
well thank you!
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
Did you get the answer?
 2 years ago

Mimi_x3 Group TitleBest ResponseYou've already chosen the best response.1
haven't tried it; looks complicated than i thought, might try it later. i already have the solution but the method looked complicated; so i tried posting it here if there is an easier method; i guess not..oh wells.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
hmm i don't think my equations are wrong but i am still suspecting something's not right with my solution. i wish angle's were easy to calculate (arctan1/3 and 63*26')
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
i hope dumbcow comes up with much easier solution.
 2 years ago

Mimi_x3 Group TitleBest ResponseYou've already chosen the best response.1
well the solution that i have uses the sum of two roots.. oh forget it
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
case 1) dw:1344008942118:dw case 2)dw:1344009010791:dw
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
assume that case 1) arrives at the point in time 't1' and case 2) arrives at time 't2' equate those two cases, dw:1344009086029:dw Two unknowns ... two equations ... solve for t1 and t2 ... (find at least one value). and put those in your equation to get the coordinate of x and y ... now you have coordinates ... find the angle of elevation.
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
@Jemurray3 is there an easier way to do this?
 2 years ago

Jemurray3 Group TitleBest ResponseYou've already chosen the best response.2
Yes, give me just a second.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
the first will give you the value of time ... use that time in to find out the coordinates ... then find the elevation as in second.
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
1)http://www.wolframalpha.com/input/?i=solve+%28x+cos+45%29+%3D+%28y+cos+63.26%29%2C+%28a+sin+63.26+x++1%2F2+9.8*+x^2%29+%3D+%28a+sin+45+y++1%2F2+9.8*+y^2%29 2) http://www.wolframalpha.com/input/?i=arctan%28%28sin+45+0.0302971++1%2F2+*+9.8*0.0302971^2%29%2F%28cos%2845%29*0.0302971%29%29
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
\[\href{http://www.wolframalpha.com/input/?i=%280.0476134%29%2F%280.0302971%29}{\text{y/x should be tan, right?}}\]dw:1344012709905:dw
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
why isn't link styling working?
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
yep ... it's close. moreover i used lots of approximations ... and got pretty close. i guess there should be better solution that would give accurate solution ... BTW what is the value of g?
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
With g=10. http://www.wolframalpha.com/input/?i=%28.0466611%29%2F%28.0296912%29 http://www.wolframalpha.com/input/?i=solve+%28x+cos+45%29+%3D+%28y+cos+63.26%29%2C+%28a+sin+63.26+x++1%2F2+10*+x%5E2%29+%3D+%28a+sin+45+y++1%2F2+10*+y%5E2%29 It still isn't 1/3. Is the question wrong?
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
well .. i guess \( 1.57 \approx 1.5 \)
 2 years ago

Jemurray3 Group TitleBest ResponseYou've already chosen the best response.2
This needs to be true for all velocities and accelerations so for convenience let's go ahead and set the velocity to 1 and the acceleration to 2. That yields the following: \[x = t\cos(\theta)\] \[y = t\sin(\theta)t^2\] Solving the first equation for t and plugging it in to the second, \[ y = x\tan(\theta)  x^2\sec^2(\theta) \] We are asked to show that \[y = \frac{1}{3} x\] so we need to show that \[x\left( \tan(\theta)  \frac{1}{3} \right)  x^2\sec^2(\theta) = 0\] for the same point x and the two different angles above. This problem is just the intersection of two parabolas. Dividing out an x and solving yields \[x = \frac{\tan(\theta)  \frac{1}{3}}{\sec^2(\theta)} \] just plug in and verify that this quantity is the same for both angles (it is).
 2 years ago

Jemurray3 Group TitleBest ResponseYou've already chosen the best response.2
You could do it with v and g left as unspecified constants but in the end you would just get the above expression multiplied by 2v^2/g ,which would still remain unchanged.
 2 years ago

Jemurray3 Group TitleBest ResponseYou've already chosen the best response.2
Lastly, if you would prefer to pretend that you didn't know the answer ahead of time, you could substitute some variable alpha rather than 1/3, and solve like this: http://www.wolframalpha.com/input/?i=%28+tan%28x%29+alpha%29+%2F+sec^2%28x%29+%3D+%28+tan%28y%29++alpha%29+%2F+sec^2%28y%29+where+x%3D+45+degrees+and+y+%3D+63+degrees+26+arc+minutes
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
lol what have i been doing ... @Ishaan94 that you are taking is ratio of time ... you need to plugin into x(t) and y(t) to get the value http://www.wolframalpha.com/input/?i=arctan%28%28sin+45+0.0302971++1%2F2+*+9.8*0.0302971^2%29%2F%28cos%2845%29*0.0302971%29%29
 2 years ago

Ishaan94 Group TitleBest ResponseYou've already chosen the best response.1
Oh, so you did get the answer @experimentX eyeopening solution @Jemurray3
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
sorry ... i forgot the answer was 1/3
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
* 1/3 = 0.3333
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.1
sorry for earlier ... i had been putting angles oppositely http://www.wolframalpha.com/input/?i=solve+%28x+cos+45%29+%3D+%28y+cos+63.26%29%2C+%28a+sin+45+x++1%2F2+9.8*+x%5E2%29+%3D+%28a+sin+63.26+y++1%2F2+9.8*+y%5E2%29 the answer would be 1/3 http://www.wolframalpha.com/input/?i=arctan%28%28sin+45+0.0966941++1%2F2+*+9.8*0.0966941%5E2%29%2F%28cos%2845%29*0.0966941%29%29 still not quite accurate
 2 years ago
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