Path of a projectile traveling along a meridian of the earth

In summary, the conversation discusses the path of a projectile traveling along a meridian of the Earth and the components of its acceleration. The first term represents the centripetal acceleration, the second term represents the coriolis acceleration, and the third term is the z component of the centripetal acceleration.
  • #1
chaoseverlasting
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3

Homework Statement


I've been working out an example by myself from the book Adv. Engg. Mathematics, E.Kreszig, section 8.7 p441.

The path of a projectile traveling along a meridian of the Earth uniformly may be given by
[tex]\vec{r(t)}=Rcos\gamma t\vec{b} + Rsin\gamma t\vec{k}[/tex] where i,j,k are unit vectors along x,y,z axes respectively.

The vector [tex]\vec{b}[/tex] is defined as [tex]\vec{b}=cos\omega t\vec{i} +sin\omega t\vec{j}[/tex]

Differentiating [tex]\vec{r(t)}[/tex] twice to get the expression for acceleration. This comes out to be

[tex]\vec{a(t)}=-R(\gamma ^2+\omega ^2)cos(\gamma t)\vec{b} -2R\gamma sin(\gamma t)\vec{b'} - R\gamma ^2cos(\gamma t)\vec{k}[/tex].

The first term is the centripital acceleration due to the Earth and the path of the projectile, the second is the coriolis acceleration. What is the third term?
 
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  • #2
It is the z component of the centripetal acceleration.
It should have sin\gamma t.
 
  • #3
Oh. ! thank you. That was incredibly stupid of me.
 

Related to Path of a projectile traveling along a meridian of the earth

What is the path of a projectile traveling along a meridian of the earth?

The path of a projectile traveling along a meridian of the earth is a curved trajectory, following the shape of the earth's surface. It travels from the point of origin along the meridian line, which runs from the North Pole to the South Pole.

What factors affect the path of a projectile traveling along a meridian of the earth?

The path of a projectile traveling along a meridian of the earth is affected by several factors, including the initial velocity, the angle of projection, air resistance, and the rotation of the earth. These factors determine the shape and distance of the projectile's path.

How does the rotation of the earth affect the path of a projectile traveling along a meridian?

The rotation of the earth causes a curved path for a projectile traveling along a meridian. This is because the earth is rotating underneath the projectile, causing it to have a curved trajectory rather than a straight line.

What is the significance of a projectile traveling along a meridian of the earth?

A projectile traveling along a meridian of the earth has important applications in fields such as physics, engineering, and navigation. Understanding its path can help in designing efficient flight paths for rockets, missiles, and satellites, as well as predicting the trajectory of objects launched from the earth's surface.

How does the curvature of the earth affect the path of a projectile traveling along a meridian?

The curvature of the earth plays a significant role in the path of a projectile traveling along a meridian. Due to the earth's curvature, the projectile will experience a gradual change in its trajectory, resulting in a curved path rather than a straight line. This curvature also affects the range and height of the projectile's path.

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