Two cars, both of mass m, collide and stick together. Prior to the collision, on

Question

Two cars, both of mass m, collide and stick together. Prior to the collision, one car had been traveling north at speed 2v, while the second was traveling at speed v at an angle phi south of east (as indicated in the figure). After the collision, the two-car system travels at speed v_final at an angle theta east of north. What is the angle theta with respect to north made by the velocity vector of the two cars after the collision? Express your answer in terms of phi. Your answer should contain an inverse trigonometric function using the notation asin, atan etc. and not arcsin, arctan etc.

Explanation / Answer

the total momentum before the impact is the same as that after the impact, in the horizontal and the vertical direction. in the horizontal direction to the right it is = m*v*cos(phi) thus the velocity in the horizontal direction after collision = m*v*cos(phi)/(m+m) ; since the total mass after the collision is m+m. in the vertically upward direction, the momentum is = m*2v - m*v*sin(phi) thus the velocity in the vertically upward direction after the collision is = ( m*2v - m*v*sin(phi) )/(m + m) thus theta = taninverse(horizontal velocity component/vertical velocity component) =taninverse( cos(phi)/(2 - sin(phi) ) )

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