2-D Collision and Explosion You are analyzing the possible defenses for an aster

Question

2-D Collision and Explosion

You are analyzing the possible defenses for an asteroid that is going to crash into the earth.
Initially, the asteroid of mass ma is traveling to the left with speed vai . A missile of mass mm with initial speed vmi to the right collides with the asteroid, and embedds itself inside the asteroid. See the figure below.

Find the common final velocity of the missile and asteroid after the collision.
Use an x-axis with positive pointing to the right.
Solve algebraically first, then use the following values for the parameters to get a value for vf x :

ma = 9.1000E+9 kg
mm = 8.1900E+8 kg
vai x = -2000 m/s
vmi x = 5555 m/s

vf x = -1.37619×103 m/s

Solve algebraically for the final speeds of pieces 1 and 2 in terms of the masses (m1 and m2), the angles, the mass before the explosion, and vf (the velocity after the collision that you found above).

Then use the following values for the parameters to get values for the speeds:

m1 = 4.9595E+9 kg
m2 = 4.9595E+9 kg
1 = 38.0 degrees
2 = 54.8 degrees

v1 =
v2 =

How much energy was released in the explosion?
Ereleased =  

Explanation / Answer

from momentum conservation

momentum before collision = momentum after collision

ma*vaix + mm*vmix = (ma+mm)*vfx

-(9.1*10^9*2000)+(8.19*10^8*5555) = ((9.1*10^9)+(8.19*10^8))*vfx

vfx = -1376.19 m/s <<<-------answer

++++++++++++++++++++++++++

from momentum conservation

momentum before collision = momentum after collision

along horizantal

ma*vaix + mm*vmix = -(m1*v1*sintheta) - (m2*v2*costheta2)

-(9.1*10^9*2000)+(8.19*10^8*5555) = -(4.9595*10^9*v1*sin38)-(4.9595*10^9*v2cos54.8).............(1)

along vertical

ma*vaiy + mm*vmiy = (m1*v1*costheta) - (m2*v2*sintheta2)

0 = (4.9595*10^9*v1*cos38)-(4.9595*10^9*v2*sin54.8).........(2)

solving 1 & 2

v1 = 2349.4 m/s

v2 = 2265.6 m/s

Ei = 0.5*Ma*vaix^2 + 0.5*mm*vmix^2

Ei = (0.5*9.1*10^9*2000^2)+(0.5*8.19*10^8*5555^2)

Ei = 3.084*10^16 J

Ef = 0.5*m1*v1^2 + 0.5*m2*v2^2

Ef = ((0.5*4.9595*10^9*2265.6^2)+(0.5*4.9595*10^9*2349.4^2)) = 2.64*10^16 J

+++++++++

energy released

E = Ei - Ef = (3.084*10^16)-(2.64*10^16) = 4.44*10^15 J

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