What's the range of projectile in this task?

[kolahalb]

A projectile of mass m is fired from the surface of the earth of radius R at an angle α from the vertical with initial speed equal to (1/[sqrt 2]) the escape velocity.How high does the projectile rise?What is its range on earth's surface?Neglect air resistance and earth's rotation.

I solved the first part easily by using work-energy theorem.The answer is 2R/3.I hope it is correct.

But I am stuck to solve the 2nd part.Will the normal formulas for the projectile problem give the answer?

 

[bulx]

projectile problem

does it mean curvature of the earth has to be considered?

 

[kolahalb]

Re: projectile problem

Yeah...
The first part was wrong.

 

[marko_benigno]

Re: projectile problem

first, solve for the x-componet (Vox)and y-component of(Voy) the initial velocity.

Voy =(1/√2)*earth esc velocity )*sinα
Vox =(1/√2)*earth esc velocity )*cosα

then solve for the time it takes for the projectile in the air. ( Vfinal (Vf) here is zero)

so,

t = (Voy-Vfy)/g
= Voy/g

then,

solve for the deltaY to find the max height to find the rise. Note: Initial Yo =0.

So, use,

deltaY= Voy*t - (1/2)*gt². the t here is the time computed earlier.

Now that we get the deltaY, its also the max rise since the inital Yo =0.

Now, to solve for the range it travelled,

deltaX = Vox*t, where Vox isthe x-component of the initial velocity.


Hope this helps.

 

[kolahalb]

Re: projectile problem

Itis wrong.
I have already done it.
You have to consider it as a central force problem.

 

[marko_benigno]

Re: projectile problem

maybe you could provide the details... what I presented was the way in solving projectile problems.... and that was what you stated in the subject heading. anyways, good for you that you got the correct answer.

 

[normaldist]

Re: projectile problem

The problem does not invlve much math if you think about it. First you must make the assumption that the mass of the earth is much greater than the projectile.

Then the projectile simply follows an ellipse with the earth at the center. The height of the projectile will be angle dependant. If the angle from vertical is 0 then the projectile will achieve a height of 2*radius of the earth. The range on the surface of the earth in angle will be 2*angle from vertical of the projectile. Therefore if the angle from vertical is 0 the range will be 0 (lands where it was shot from), if the angle from vertical is 90 the range will be 180 degrees of the earths surface (the other side).

Note: If the angle of departure is not zero then the height < 2*radius of the earth because the projectile will have some kinetic energy at the apex of flight.

C

 

[Mr.MEB]

projectile problem

it is the projectile problem which as variable g with the altitude. You must replace g with g(h)=...
You will also need to know the earth's mass, but that can be found with escape velocity. Try to think with polar coordinate, forget about x.