31 Asteroid Collision. Two asteroids of equal mass in the asteroid belt between

Question

31 Asteroid Collision. Two asteroids of equal mass in the asteroid belt between Mars and Jupiter collide with a glancing blow. Asteroid A, which was initially traveling at 40.0 m/s, is de- flected 30.0° from its original direction, while asteroid B, which was initially at rest, travels at 45.0 to the original direction of A (Fig. E31). (a) Find the speed of each asteroid after the collision. (b) What fraction of the original kinetic energy of asteroid A dissipates during this collision? Figure E31 40.0 m/s 30.0° 45.0°

Explanation / Answer

here,

masses of asteroids be m

let the final speeds be v1 and v2

using conservation of momentum

m * 40 m/s i = m * v1 + m * v2

40 i = v1 * ( cos(30) i + sin(30) j) + v2 * ( cos(45) i - sin(45) j)

on compairing

0.866 * v1 + 0.707 * v2 = 40 ....(1)

and

0.5 * v1 - 0.707 * v2 = 0 ....(2)

from (1) and (2)

v1 = 29.3 m/s

v2 = 20.7 m/s

b)

the initial kinetic energy , KEi = 0.5 * m * 40^2 = 800 *m

the final kinetic energy , KEf = 0.5 * m * v1^2 + 0.5 * m * v2^2

KEf =0.5 * m * ( 29.3^2 + 20.7^2) = 643.49 * m

the fraction of original kinetic energy dissapated in the collison , f = (KEi - KEf)/KEi

f = ( 800 * m - 634.49 *m)/(800 * m)

f = 0.2

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