- #1
sankalpmittal
- 785
- 25
Here's a equation of a projectile thrown in a parabolic path :
y=xtanθ - gx2/2u2cos2θ
where x is the corresponding x - coordinate or the range of a projectile at a point ,
y is the corresponding y - coordinate or the height of projectile at point,
θ is the angle made with the horizontal arbitrary x-axis through which projectile is thrown.
u is magnitude of initial velocity vector making angle θ with horizontal.
Note: Point from where the projectile began is assumed to be origin , i.e. (0,0) coordinate.
I was once told that in order to solve difficult problems of projectile , calculus is a very powerful tool.
We can consider y in equation as a function of θ and then can differentiate y with respect to θ {find y'(θ)}.
dy/dθ = ...
Then we can put dy/dθ =0 and also dx/dθ =0. Hence we can easily solve for optimal angle θo for maximum range. Sometimes we just put dy/dθ =0 when we are given "x" and we can solve for θ.
Also if we imagine the equation to be plotted on a graph then we can obviously consider y as a function of x and then differentiate y with respect to x {find y'(x)}.
Then we put dy/dx=0 to find point of maxima or minima , I think. We also find d2y/dx2.
So here are my questions :
What's the use or when do we find dy/dθ or dy/dx in the projectile equation? I cannot understand. Also I cannot fathom this all theoretically. Why are we doing such thing ? Can someone explain ? Is this really a powerful tool to solve projectile problems ? When do we put dy/dθ =0 and also dx/dθ =0 or even dy/dx =0 and why , while differentiating w.r.t θ or x in y=xtanθ - gx2/2u2cos2θ ?
y=xtanθ - gx2/2u2cos2θ
where x is the corresponding x - coordinate or the range of a projectile at a point ,
y is the corresponding y - coordinate or the height of projectile at point,
θ is the angle made with the horizontal arbitrary x-axis through which projectile is thrown.
u is magnitude of initial velocity vector making angle θ with horizontal.
Note: Point from where the projectile began is assumed to be origin , i.e. (0,0) coordinate.
I was once told that in order to solve difficult problems of projectile , calculus is a very powerful tool.
We can consider y in equation as a function of θ and then can differentiate y with respect to θ {find y'(θ)}.
dy/dθ = ...
Then we can put dy/dθ =0 and also dx/dθ =0. Hence we can easily solve for optimal angle θo for maximum range. Sometimes we just put dy/dθ =0 when we are given "x" and we can solve for θ.
Also if we imagine the equation to be plotted on a graph then we can obviously consider y as a function of x and then differentiate y with respect to x {find y'(x)}.
Then we put dy/dx=0 to find point of maxima or minima , I think. We also find d2y/dx2.
So here are my questions :
What's the use or when do we find dy/dθ or dy/dx in the projectile equation? I cannot understand. Also I cannot fathom this all theoretically. Why are we doing such thing ? Can someone explain ? Is this really a powerful tool to solve projectile problems ? When do we put dy/dθ =0 and also dx/dθ =0 or even dy/dx =0 and why , while differentiating w.r.t θ or x in y=xtanθ - gx2/2u2cos2θ ?
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