Which Regions Can This Cannon Reach with Its Projectile?

In summary, the conversation was about finding the region of space that a projectile can reach when shot from a cannon with an initial velocity and direction. The formula y=x tan alpha - gx^2/v_0^2 (tan^2 alpha -1) was derived and it was discussed that this is a quadratic equation. One person, Jaan Kalda, suggested adding y+gx^2/2v_0^2 to the equation, but the reasoning behind it was not fully understood.
  • #1
roborangers
3
1
Homework Statement
A cannon is situated in the origin of coordinate axes
and can give initial velocity v0 to a projectile, the shooting direction can be chosen at will. What is the region of space R
that the projectile can reach?
Relevant Equations
but when i checked the solution i say that kalda added y+gx^2/2v_0^2 but i dont understand why
what i tried to do is to write y=v_0tsin alpha - 1/2gt^2 and x=v_0 cos alpha tand that t=x/v_0 cos alphai plug t in the formula for y and get that y= x tan alpha - gx^2/v_0^2 (tan^2 alpha -1)since jaan klada said there should be a quadratic equation (because its a parabola) i thought that gx^2/v_0^2 tan^2 alpha is a, -x tan alpha is b and gx^2/2v_0 is c and got another formula
 
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  • #2
roborangers said:
Homework Statement: A cannon is situated in the origin of coordinate axes
and can give initial velocity v0 to a projectile, the shooting direction can be chosen at will. What is the region of space R
that the projectile can reach?
Relevant Equations: but when i checked the solution i say that kalda added y+gx^2/2v_0^2 but i dont understand why

what i tried to do is to write y=v_0tsin alpha - 1/2gt^2 and x=v_0 cos alpha tand that t=x/v_0 cos alphai plug t in the formula for y and get that y= x tan alpha - gx^2/v_0^2 (tan^2 alpha -1)since jaan klada said there should be a quadratic equation (because its a parabola) i thought that gx^2/v_0^2 tan^2 alpha is a, -x tan alpha is b and gx^2/2v_0 is c and got another formula
This is not easy to read. Punctuation and spacing are important.
 
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  • #3
PeroK said:
This is not easy to read. Punctuation and spacing are important.
yes you are righ but i got it
 
  • #4
The correct equation for the projectile trajectory is $$y=x\tan\alpha-\frac{gx^2}{2g}(1+\tan^2\alpha).$$The general equation for the quadratic equation is $$ax^2+bx+c=0$$.What exactly is your question? When you say "What is the region of space R that the projectile can reach?" do you mean in the horizontal direction only or in two dimensional space?

I don't know who Jaan Kalda is, but I think that you should include the whole answer that he provided not just the term that he added.
 
  • #5
yes exactly i got that y is v_0^2/2g - gx^2/2v_0^2
 

FAQ: Which Regions Can This Cannon Reach with Its Projectile?

What is the Jaan Kalda Kinematics question about?

The Jaan Kalda Kinematics question explores the regions that a cannon can reach with its projectile, considering factors like initial velocity, angle of projection, and gravitational forces. It involves determining the set of points or regions in space that the projectile can reach under given conditions.

What assumptions are made in solving this kinematics problem?

Common assumptions include neglecting air resistance, assuming constant gravitational acceleration, and considering the projectile motion in a vacuum. These simplifications allow for the application of basic kinematic equations to determine the trajectory and reachable regions.

How do you determine the maximum range of the projectile?

The maximum range of the projectile is determined using the formula \( R = \frac{v_0^2 \sin(2\theta)}{g} \), where \( v_0 \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity. The range is maximized when the angle of projection is 45 degrees.

What factors influence the shape of the reachable region?

The shape of the reachable region is influenced by the initial velocity of the projectile, the angle of projection, and the gravitational acceleration. Varying these parameters changes the trajectory, altering the set of points that the projectile can reach.

Can this problem be solved analytically or does it require numerical methods?

This problem can often be solved analytically using kinematic equations for projectile motion. However, for more complex scenarios, such as those involving varying gravitational fields or air resistance, numerical methods may be required to accurately determine the reachable regions.

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