$.31..$ 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 $40\mathrm{.}0\frac{m}{s},$ is deflected $30\mathrm{.}0$ from its original direction, while asteroid $B,$ which was initially at rest, travels at $45\mathrm{.}0$ to the original direction of $A($Fig$.$ E$8\mathrm{.}31).($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 E$8\mathrm{.}31$