# Can't figure out the math

#### Wizardsblade

Hey, I was trying doing some calculations and ran across a question I just can’t figure out. Hopefully I explain this well.
I am trying to find the Z component of velocity of an object on earth’s surface after a given time. For example a rock lies on the ground with a tangential velocity of ~450m/s (I believe). 1 second later it will be going v1 speed in the negative z direction and v2 speed in the x direction. I believe v1^2+v2^2=450^2 but I really don't know where to go from here.

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#### Tide

Science Advisor
Homework Helper
Define your coordinate system and state the problem a little more carefully. At it stands it is unclear.

#### Wizardsblade

At time = 0 the z axis is up and down and the x axis is forward and backward in the direction of earths rotation, i.e. looking at the earth form above the earth. What I am trying to figure out is the x and z components at any give time. For example at time = 6 hours the x component will be 0 and the z component will be -450m/s, at time = 12 hours z=0 and x=-450m/s.

#### Wizardsblade

Wait I think I figured out a way. I did this: t=1s so I figured the number of seconds in a day (24*60*60=86400) and divided 360 degrees by 86400 and got .0041. Then I took the sin of .0041 and multiplied by 450m/s. So I got 7.2x10^-5m/s. Is this correct for the z component?

#### Tide

Science Advisor
Homework Helper
Unless you're standing at the equator you're going to need three basis vectors in the fixed frame of reference to describe those vectors.

What you are calling z corresponds to the radial coordinate in a spherical coordinate system so that a unit vector in the direction is given by

$$\hat r = \sin \theta \cos \phi \hat i + \sin \theta \sin \phi \hat j + \cos \theta \hat k$$

With the simple rotation you've defined simply replace $\phi$ with $\omega t$. Also, what you are calling the x direction corresponds to a vector in the azimuthal direction ($\phi$) and the corresponding unit vector is given by

$$\hat \phi = -\sin \phi \hat i + \cos \phi \hat j$$

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