Angle required to hit the target in projectile motion

In summary, the conversation discusses the problem of projectile motion and the final formula for theta, as well as the equation (2) which represents a relationship between x, y, g, and v that results in a single solution for theta. It is suggested that this equation represents the locus of points in the plane that can be reached by firing the projectile at a specific angle.
  • #1
Soren4
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2
Consider the problem of projectile motion where the angle to hit a target ##(x,y)## is asked, once given the initial velocity magnitude ##v_0##. The projectile is fired from the point ##(0,0)##.

Here is the final formula for theta, solving the equations of motion (https://en.wikipedia.org/wiki/Traje...#Angle_required_to_hit_coordinate_.28x.2Cy.29)

$$\theta = \arctan{\left(\frac{v_0^2\pm\sqrt{v_0^4-g(gx^2+2yv_0^2)}}{gx}\right)} \tag{1}$$

Now if I impose ##\Delta=v_0^4-g(gx^2+2yv_0^2)## (for which we have the only solution ##\theta=\arctan \frac{v_0^2}{gx}##) I find $$y=\frac{v_0^2}{2g}-\frac{g}{2v_0^2}x^2\tag{2}$$

Now in my view this is the equation of a parabola which describes the points ##(x,y)## in plane that can be hit only firing at ##\theta=\arctan \frac{v_0^2}{gx}##, once given the initial velocity ##v_0##.

On my textbook it is claimed that ##(2)## represents the equation of the trajectory of the projectile, if fired at ##\theta=\arctan \frac{v_0^2}{gx}##.

I don't think that this is possible, to begin with the fact that ##(2)## does not pass through ##(0,0)##.

Can anyone tell me what exactly ##(2)## means?
 
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  • #2
(2) is a relationship between x, y, g and v that results in a single solution for theta.
 
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  • #3
PeroK said:
(2) is a relationship between x, y, g and v that results in a single solution for theta.

Thanks for the reply! Ok that's a relation between ##x,y,v##. Is it correct to see it as the locus of points in plane that, given the particular value of ##v##, can be reached iff the projectile is fired at the angle ##\theta=\arctan \frac{v_0^2}{gx}##? For those points there is infact this one only solution in ##(1)##.
 

Related to Angle required to hit the target in projectile motion

1. What is projectile motion?

Projectile motion refers to the motion of an object that is launched into the air and moves along a curved path under the influence of gravity.

2. How is the angle required to hit a target in projectile motion calculated?

The angle required to hit a target in projectile motion is calculated using the equation: θ = tan-1(vy/vx), where θ is the angle, vy is the vertical component of velocity, and vx is the horizontal component of velocity.

3. What factors affect the angle required to hit a target in projectile motion?

The angle required to hit a target in projectile motion is affected by the initial velocity, the angle of launch, the acceleration due to gravity, and the distance to the target.

4. How does air resistance affect the angle required to hit a target in projectile motion?

Air resistance can slightly alter the angle required to hit a target in projectile motion, as it can slow down the object and cause it to follow a slightly different trajectory. However, the effect of air resistance is usually minimal and can be ignored in most cases.

5. Can the angle required to hit a target in projectile motion be negative?

Yes, the angle required to hit a target in projectile motion can be negative. This indicates that the target is below the initial position of the object, and the object must be launched at an angle below the horizontal in order to hit the target.

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