User:Oh Isaac/R6-10

Given:
The schematic of the motion of a particle in the air :

Derive the equations of motion:

Consider the case $$k\neq 0$$ and $$v_{x0}=0$$: ($$) turns into

Which can be written as :

Separation of variables

For all values of $$n$$ :

Give particular values of some parameters :

Initial vertical velocity:$$z(t):=v_y(0)=50$$

Find:

 * For each value of $$n$$, find the vertical velocity $$z(t)$$ vs. time t, plot this function.
 * For each value of $$n$$, find the altitude $$y(t)$$ vs. time t, find the time when the projectile returns to the ground.

Use numerical methods to find the value of each given value of time t if the explicit expression cannot be obtained.


 * 1) $$n=2$$
 * 2) $$n=3$$

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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According to $$, $$and$$, the uniform expression can be written as:

The inverse funciton of $$can be show as below in terms of $$t$$:

So we get:

When n=2
The $$-b come to be:

With the initial value$$-c:

The accurate number of $$k$$ is $$\frac{1536841641999493}{4503599627370496}$$ byWolframApha and WolframApha, and the form of $$can be written as:

So we can plot the figure of z vs t as follow:

The inverse function of $$can be obtained by Matlab command $$ g = finverse(f,var) $$as follows :

So we can plot the figure of t vs z as follow:

Integrate as $$shows we get:

when the projectile reach the ground, y, the altitude, reach 0. So we need to solve the t in this equation below:

UseWolframAlpha to get the answer. The answer is

We can see that the expression of y is a  periodic function, the positive value which is nearest to 0 is $$0.6825$$, at which time the projectile reach the ground.

So can we solve this problem by numerical methods. The program and plot obtained by MATLAB are showed below

When n=3
The $$-b come to be:

With the initial value$$-c:

The number of $$k\doteq0.20667022059582652141$$ by WolframAlpha, so we just write the expression as below :

So we can plot the figure of z vs t as follow:

The inverse function of $$cannot be obtained by MATLAB, So numerical methods is needed.

Discard the part that contains imaginary number for these have no physical meaning, then exchange the axes of z vs t to t vs z, get the graph like this:

Then integrate z to get y. When the projectile reach ground, the value of y should reach 0, which means at the t in which the area of blue and red are equal in $$.The plot and MATLAB programs are showed below.

The answer, according to the $$, is $$0.5024$$, at which time the projectile reach the ground.

*When n=1
The $$-b come to be:

With the initial value$$-c:

The number of $$k=\frac{\log{11}}{2}$$ by WolframAlpha, so we just write the expression as below :

So we can plot the figure of z vs t as follow. Discard the part that contains imaginary number afterwards for these have no physical meaning:

The inverse function of $$showed below can be obtained by MATLAB, and so can numerical methods.

Then integrate z to get y. When the projectile reach ground, the value of y should reach 0, which means at the t in which the area of blue and red are equal in $$.

We can get the answer is$$5.49991$$, at which time the projectile reach the ground, from WolframAlpha. And so can we get it from numerical methods.The plot and MATLAB programs are showed below.