Projectile motion theta question

[Ara macao]

A projectile is launched from origin at angle theta with horizontal, whose position is given by r(t). For small angles, distance from origin always increases. But if projectile is launched nearly straight up, it goes to a farthest point and then moves back down towards origin, so distance to origin first increases, then decreases. Which initial launch angle theta divides the two types of motion?

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I set up an equation of x^2+y^2 where x = vcostheta and y = vsintheta - 4.9x^2 and then took the derivative of it with respect to t, setting it equal to 0. However, that gets really messy and is there a better way to do it? I'm not sure about my answer anyways. Can someone please guide me though the process? Thanks.

 

[Päällikkö]

I did manage to get an answer by doing the same things as you did (derivating and setting that 0).

It was surprisingly short, even though it seemed quite long at the beginning. I got 71 degrees, which seems somewhat logical (I'm telling this as you don't seem to have the correct answers in the book).

 

[quicknote]

I would recommend using the principles of projectile motion to analyze this situation. The initial launch angle theta that divides the two types of motion can be determined by looking at the vertical component of the projectile's velocity.

For small angles, the vertical component of the velocity is always positive, meaning the projectile will continue to increase in height. However, as the angle theta increases, the vertical component of the velocity will eventually become 0 at the highest point of the projectile's trajectory. At this point, the projectile will begin to fall back towards the origin, resulting in a decrease in distance from the origin.

To determine the exact angle theta that divides these two types of motion, we can use the equation for the maximum height of a projectile, which is h = (v^2sin^2(theta))/2g, where v is the initial velocity and g is the acceleration due to gravity.

If we set the equation equal to 0, we can solve for theta and find that the angle theta that divides the two types of motion is 45 degrees. Therefore, any initial launch angle below 45 degrees will result in a continuously increasing distance from the origin, while any angle above 45 degrees will result in a maximum distance from the origin before falling back towards it.

In terms of the approach you mentioned, setting up an equation of x^2+y^2 and taking the derivative, this can also be a valid method. However, it may be more complicated and require more mathematical understanding. Using the principles of projectile motion is a more straightforward approach. I hope this helps guide you through the process.