So, what, I just take the first one?Yes, take the first one.

In summary, the conversation discusses using a cannon with a muzzle speed of 1000 m/s to destroy a target 2000m away and 800m above the ground. The problem involves finding the angle at which the cannon should be fired, ignoring air friction. The conversation goes through the equations and calculations necessary to solve the problem, including setting up a quadratic equation and using the quadratic formula to find the angle. The final answer is approximately 21.241 degrees.
  • #1
neutron star
78
1
Now what? Quadratic equation!

Homework Statement


A cannon having a muzzle speed of 1000 m/s is used to destroy a target on a mountaintop. The target is 2000m from the cannon horizontally and 800m above the ground. At what angle, relative to the ground, should the cannon be fired? Ignore air friction.


Homework Equations





The Attempt at a Solution


Xf=Xo+Vox t = Xo+Vo cosΘt
Yf=Yo+Voy t -1/2gt[tex]^2[/tex] = -Yo+VosinΘt-1/2gt[tex]^2[/tex]

t=2000/1000cosΘ = 2/cosΘ

800=1000sinΘ(2/cosΘ) - 1/2g (2/cosΘ)[tex]^2[/tex]
=2000tanΘ-1/2g 4/cos[tex]^2[/tex]Θ
1/cos[tex]^2[/tex]-sec[tex]^2[/tex]=1+tan[tex]^2[/tex]

800=tanΘ-2g(1+tan[tex]^2[/tex]Θ)

ax[tex]^2[/tex]+bx+c=0

Ok, what do I do to get the angle now, grr I'm drawing a blank, I don't have long to finish this :\.
 
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  • #2


Set [tex]\tan{\theta}\equiv Z[/tex]

You know have a quadratic equation in [tex]Z[/tex]
 
  • #3


19.62z[tex]^2[/tex]+2000z-780.38

or

20z[tex]^2[/tex]+2000z-780

But now what?
 
  • #4


neutron star said:
19.62z[tex]^2[/tex]+2000z-780.38

or

20z[tex]^2[/tex]+2000z-780

But now what?

Don't round those off. That's a very bad idea, especially since you're going to deal with [tex]\tan{\theta}[/tex] soon and that function is sensitive to such small changes.

You have a quadratic equation, how do you solve a quadratic equation?
 
  • #5


RoyalCat said:
Don't round those off. That's a very bad idea, especially since you're going to deal with [tex]\tan{\theta}[/tex] soon and that function is sensitive to such small changes.

You have a quadratic equation, how do you solve a quadratic equation?

-b+or- sq root b^2-4ac all over 2a

I did that and got weird answers.
 
  • #6


I got x=-15.37 x=-3984.63
 
  • #7


neutron star said:
I got x=-15.37 x=-3984.63

You plugged your numbers in wrong.

I got:

[tex]Z_1\approx 0.3887[/tex]

This correlates to: [tex]\theta=21.241^o[/tex]

[tex]Z_2\approx-102.3255[/tex]

This correlates to: [tex]\theta=-89.44^o[/tex] which is utter nonsense.
 

1. What is a quadratic equation?

A quadratic equation is a mathematical equation that contains terms of the form ax^2 + bx + c, where a, b, and c are constants and x is the variable. It is used to solve problems involving parabolas and can have two solutions.

2. How do I solve a quadratic equation?

There are multiple ways to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. The most commonly used method is the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by x = (-b ± √(b^2-4ac)) / 2a.

3. What is the discriminant and how is it used in quadratic equations?

The discriminant is the part of the quadratic formula inside the square root, b^2-4ac. It is used to determine the number of solutions to a quadratic equation. If the discriminant is positive, there will be two real solutions. If it is zero, there will be one real solution. And if it is negative, there will be no real solutions.

4. Can a quadratic equation have complex solutions?

Yes, a quadratic equation can have complex solutions. This occurs when the discriminant is negative, meaning there are no real solutions. In this case, the solutions will be complex numbers of the form a+bi, where a and b are real numbers and i is the imaginary unit (√-1).

5. Why are quadratic equations important?

Quadratic equations have many real-world applications in physics, engineering, and other sciences. They are used to solve problems involving projectile motion, optimization, and finding the roots of polynomials. They also provide a basis for understanding more complex mathematical concepts, such as graphing and trigonometry.

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