Rotational earth motion I am extremely close to what the answer is supposed to be

In summary, the homework statement states that if a particle is projected vertically upward at a height h above the Earth's surface at a northern latitude θ, it will strike the ground at a point 4/3*ωcosθ*sqrt(8h^3/g) to the west. The attempt at a solution first reduces the equation to find x' and z', then uses these values to solve for t.
  • #1
jb646
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0

Homework Statement



If a particle is projected vertically upward to a height h above the Earth's surface at a northern latitude θ, show that it strikes the ground at a point 4/3*ωcosθ*sqrt(8h^3/g) to the west. (Neglect air resistance and consider only small vertical heights.)

my initial conditions as I see them (...I have nothing more than I have already written):
I believe that (x0',y0',z0')=(0,0,h) and (x0dot',y0dot',z0dot')=(0,0,0)

the notation is '=in a spinning situation while dot=derivative so my initial position should be as I wrote (00h) and my initial velocity (000)

Homework Equations



x'(t)=1/3*w*g*t^3*cos(θ)-w*t^2(z0dot'*cos(θ)-y0dot'*sin(θ))+x0t+x0'

y'dot(t)=y0dot'(t)-w*xdot'*t^2*sin(θ)-2wtsinθ*x0'+y0'

z'(t)=-1/2gt^2+z0dot*t+wx0dot'*t^2cosθ+2wx0'tcosθ+z0

The Attempt at a Solution



reducing using my initial conditions, I believe that:

x'=1/3wgt^3cosθ and z'=-(gt^2)/2+h which setting z'=0 gives me t=sqrt(2h/g)

I then plugged t into x':
x'=1/3wsqrt(8h^3/g)cosθ but it is supposed to be 4/3, not 1/3 as is stated in the problem

Is one part of my initial equation for x'(t) supposed to be wgt^3cosθ or did I mess up my algebra somewhere
 
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  • #2


Hey, I can't really understand all the mess is in the middle. I'm just going to use prime' for time derivative because I don't want to use latex. I made my coordinate system as z normal to the earth, y tangent to the earth, and x out of the page.

We know that the coriolis force is wxv, which gives only the force in the x direction
Fc=-2z'cosØ xhat
z''=-g
z'=v0-gt
z=v0t-1/2 gt^2
Since I hate mucking around with signs, move to the left for now
-x''=2z'wcosØ
-x'=2zwcosØ + x0
-x0=0 by IC
-x'=2(v0t-1/2 gt^2)

Take it from here and you ought to get it right.
 
  • #3


hey thanks a lot, that really helped out, I have it now.
 

1. What causes the earth to rotate?

The earth rotates due to its angular momentum, which is a result of its initial spin during its formation and the force of gravity between the earth and the sun.

2. How fast does the earth rotate?

The earth rotates at a speed of about 1670 kilometers per hour at the equator. This speed decreases as you move towards the poles.

3. Is the earth's rotation constant?

No, the earth's rotation is not constant. It is gradually slowing down due to the tidal forces from the moon and the sun.

4. Does the earth rotate in a perfect circle?

No, the earth rotates in an elliptical orbit around the sun. This means that the distance between the earth and the sun varies slightly throughout the year.

5. What is the impact of the earth's rotation on our daily lives?

The earth's rotation is responsible for the day and night cycle, as well as the Coriolis effect which influences weather patterns and ocean currents. It also affects the length of a day and the direction of shadows cast by the sun.

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