Posted by **sana** on Tuesday, March 6, 2012 at 9:44am.

Consider the Sun to be at the origin of an xy-coordinate system. A telescope spots an asteroid in the xy-plane at a position given by (1.8 1011 m, 3.1 1011 m) with a velocity given by (-8.4 103 m/s, -6.3 103 m/s). What will be the magnitude of the asteroid's velocity and it's distance from the Sun at closest approach?

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**drwls**, Tuesday, March 6, 2012 at 11:14am
This is a very clever problem. Here is how to do it.

The cross product of the V and R vectors is the rate at which area is swept out by the asteroid, which (according to Kepler's second law) is a constant related to the angular momentum. Compute its value using the experimental measurements.

C1 = |R|*Vtheta

= [(3.1)(8.4) - (6.3)(1.8)]*10^11

= 14.7*10^11 m^2/s

Vtheta is the velocity component perpendicular to the position vector, R

Another constant of motion is the total energy, which is (1/2)mV^2 -GMm/R.

Since m (the asteroid mass) is constant,

V^2/2 - GM/R = C2 , another constant. Compute this constant using G, M(solar) and the observed values of the magnitudes of R and V.

At perihelion, the C1 constant is the product of the velocity Vmax and closest approach distance, Rmin

Vmax*Rmin = C1

Vmax^2/2 - G*M/Rmin = C2

You now have two equations for the two unknowns, Vmax and Rmin. You will need to use the other known constants, M and G

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**Sana**, Tuesday, March 6, 2012 at 12:16pm
Could u rewrite the eqn that you used for C2

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**drwls**, Tuesday, March 6, 2012 at 12:37pm
(V^2/2) - (GM/R) = C2

(Vmax^2/2) - (G*M/Rmin) = C2

It is the sum of kinetic and potential energy per unit mass. Using the first equation and the |V| and |R| magnitudes of the observation, you can compute the value of C2.

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**please**, Tuesday, March 6, 2012 at 3:50pm
i am unable to get the right answer..please help

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**drwls**, Tuesday, March 6, 2012 at 4:36pm
Sorry. You'll have to fill in the blanks. I have explained how to do it. Make sure you are using the correct values of G and M, and that your equations are dimensionally consistent.

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**Anonymous**, Thursday, March 15, 2012 at 4:55pm
what drwls did makes no sense..

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