To solve this problem, we can use the law of conservation of momentum, which states that the total momentum of a system before a collision is equal to the total momentum after the collision. In this case, the momentum of the system is:
p = m1v1 + m2v2
where m1 and v1 are the mass and velocity of car A before the collision, and m2 and v2 are the mass and velocity of car B before the collision. Since car B is moving to the right after the collision, we can assume that its velocity before the collision was zero.
Before the collision, the total momentum of the system is:
p1 = m1v1 + m2v2 = (20 mg)(v1) + (15 mg)(0) = 20 mg v1
After the collision, the total momentum of the system is:
p2 = m1v1' + m2v2'
where v1' is the velocity of car A after the collision and v2' is the velocity of car B after the collision. We know that car B moves to the right with a speed of 2 m/s after the collision, so we can substitute v2' = 2 m/s into the equation:
p2 = m1v1' + m2v2' = (20 mg)(v1') + (15 mg)(2 m/s)
Since the collision is elastic and the cars rebound, we can assume that the total kinetic energy of the system is conserved. Therefore, we can set the initial kinetic energy equal to the final kinetic energy:
(1/2)m1v1^2 + (1/2)m2(0)^2 = (1/2)m1(v1')^2 + (1/2)m2(2 m/s)^2
Simplifying and solving for v1', we get:
v1' = v1 - [(2m/s)(15mg)] / (20mg + 15mg) = v1 - 0.2 m/s
We also know that the two cars are in contact for 0.5 seconds, so we can use the impulse-momentum theorem to find the average impulsive force:
F_avg = Δp / Δt
where Δp is the change in momentum and Δt is the time interval. We can calculate Δp as:
Δp = p2 - p1 = (20 mg v1') - (20 mg v1) = -4 mg(v1 - 0.2 m/s)
Since the collision time is 0.5 seconds, we can substitute Δp and Δt into the equation:
F_avg = Δp / Δt = [-4 mg(v1 - 0.2 m/s)] / (0.5 s) = -8 mg(v1 - 0.2 m/s) N
Therefore, the velocity of car A after the collision is v1' = v1 - 0.2 m/s, and the average impulsive force between the cars is -8 mg(v1 - 0.2 m/s) N. Note that the negative sign indicates that the force is acting in the opposite direction of the initial motion of car A.