For convenience later in the solution, the coordinate system is based on the impact point P. Due to the complicated geometry, it is useful to use vector notation.

At the time of impact, the position vector of each car's center of mass relative to the point of impact is given by

     r_1/p = (3 sinα - 6 cosα)i + (6 sinα + 3cosα)j ft
           = (3 sin45 - 6 cos45)i + (6 sin45 + 3cos45)j ft
           = -2.121i + 6.364j ft

     r_2/p = -5i - 3j ft

Car Velocity Vectors Before Impact

Car Velocity Vectors After Impact

Prior to impact, the velocity of each cars' center of gravity is

     v₁ = v_1xi + v_1yj = 52 cosαi - 52 sinαj
           = 36.77i - 36.77j ft/s
     v₂ = 0i + 0j
     ω₁ = 0k
     ω₂ = 0k

After the impact, the velocities are

     v'₁ = v'_1xi + v'_1yj
     v'₂ = v'_2xi + v'_2yj
     ω'₁ = ω'₁k
     ω'₂ = ω'₂k

Since friction is neglected, the force of the impact is entirely in the y direction; therefore, the x components of velocity remain unchanged,

     v'_1x = v_1x = v₁cos45 = 36.77 ft/s 
                                                                       (7)
     v'_2x = 0

Linear momentum in the y direction must be conserved giving

     v_1y m₁ = v'_1y m₁ + v'_2y m₂                           (8)

     (36.77) 2500/g = v'_1y 2500/g + v'_2y 3500/g  

     36.77 = v'_1y + 1.4 v'_2y

Since the force of the impact does not generate a moment about the impact point for either car, angular momentum is conserved for each about the impact point, P,

     I_1-cgω₁ + (r_1/p × m₁v₁)•k
                             = I_1-cgω'₁ + (r_1/p × m₁v'₁)•k     (9)

     I_2-cgω₂ + (r_2/p × m₂v₂)•k
                             = I_2-cgω'₂ + (r_2/p × m₂v'₂)•k    (10)

Applying the initial conditions to Eqs. 9 and 10 and simplifying gives (I₁ = I_1-cg and I₂ = I_2-cg)

     (r_1/p × m₁v₁)•k = I₁ω'₁ + (r_1/p × m₁v'₁)•k    

     0 = I₂ω'₂ + (r_2/p × m₂v'₂)•k

Completing the the cross products give

     m₁ (3sinα - 6cosα) v_1y
              = I₁ ω'₁ + m₁ (3sinα - 6cosα) v'_1y          (11)

     0 = I₂ ω'₂ + 5 m₂ v'_2y                                    (12)

The coefficient of restitution determines the relative velocities of the point of impact before and after the impact. The basic form is

                                           (13)

Since the velocity of point P before the impact for each car is identical to the center of gravity velocity, Eq. 13 simplifies to

                                           (14)

The velocity of the point of impact after the collision for each car can be related to the velocity of the center of gravity and the rotational velocity through the relationships

     v'_1p = v'₁ + ω'₁ × r_p/1   

     v'_2p = v'₂ + ω'₂ × r_p/2  

Substituting known values, performing the cross products, and then equating the i and j components, gives

     v'_1px = v_1x + (6sinα + 3cosα) ω'₁

     v'_1py = v'_1y + (6cosα - 3sinα) ω'₁

     v'_2px = -3ω'₂

     v'_2py = v'_2y + 5ω'₂

Substitute into Eq. 14 and simplify to get

  (15)

Equations 8, 11, 12 and 15 form a system of equations that can be solved for the four unknowns v'_1y, v'_2y, w'₁ and w'₂.

Solving this system of equations give the following four unknowns

Substitute the known values before the collision to get the results

     v'_1y = -21.8 mph
     v'_2y = -10.7 mph

     ω'₁ = 178.2 deg/s
     ω'₂ = -233.4 deg/s