Answer to Question #171707 in Physics for RHYLYN

Question #171707

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 deflected 30.0° from its original direction, while asteroid B, which was initially at rest, travels at 45.0° to the original direction of A (see figure below).

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?



1
Expert's answer
2021-03-17T16:56:21-0400

Draw a diagram:



Apply conservation of momentum for the asteroids before and after the collision:


pA+pB=pA+pB.p_A +p_B=p'_A+p'_B ​ .

Assuming asteroids were moving from left to right, wrote the conservation of momentum along each axis:


Ox:mvA=mvA cos30°+mvBcos45°,Oy:0=mvA sin30°mvB sin45°.Ox: mv_A=mv'_A\text{ cos}30°+mv'_B\text{cos}45°,\\ Oy: 0=mv'_A\text{ sin}30°-mv'_B\text{ sin}45°.

From the second equation, find how velocity of A depends on velocity of B:


vA=vB2.v'_A=v'_B\sqrt2.

Substitute this in the first equation for Ox and find final velocity of B:


vB=20.7 m/s,vA=29.3 m/s.v'_B=20.7\text{ m/s},\\ v'_A=29.3\text{ m/s}.

Kinetic energy before and after the collision:


K1=m2vA2, K2=m2vA2.K_1=\frac m2v^2_A,\\\space\\ K_2=\frac m2 v'^2_A.

Dissipated energy:


ΔK=K1K2=m2(vA2vA2). k=ΔKK1=1vA2vA2=0.46ΔK=K_1 −K_2=\frac m2(v_A^2-v'^2_A).\\\space\\ k=\frac{\Delta K}{K_1}=1-\frac{v'^2_A}{v^2_A}=0.46

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