Question
The figures below show bird's-eye views of six automobile crashes an instant before they occur. The automobiles have different masses and incoming velocities asshown. After impact, the automobiles remain joined together and skid to rest in the direction shown by vfinal. Rank these crashes according tothe angle θ , measured counterclockwise as shown, at which the wreckage initially skids.
Solution
Concepts and reason
The required concepts to solve the problem are collision and momentum conservation.First use the momentum conservation to find a general equation for angle made by the automobiles after collision.Then rank, from largest to smallest, the crashes according to the angle measured counterclockwise.
Fundamentals
Velocity is a vector with both magnitude and direction. The direction of vector depends on the angle made by the vector with respect to the horizontal or with respect to the vertical. If the angle is made with respect to the horizontal, then thexxcomponent of the vector is given by the \cos \thetacosθ component of the vector and theyycomponent of the vector is given by the \sin \thetasinθ component of the vector.According to the conservation of momentum, the momentum before collision is equal to the momentum after collision.If the collision between two objects is considered, the conservation of momentum gives the relation,{m_1}{u_1} + {m_2}{u_2} = {m_1}{v_1} + {m_2}{v_2}m1u1+m2u2=m1v1+m2v2Here, {m_1}m1is the mass of the first object, {m_2}m2is the mass of the second object, {u_1}u1is the initial velocity of the first object, {u_2}u2is the initial velocity of the second object, {v_1}v1is the velocity after collision of the first object and{v_2}v