Final answer:
The final velocities of both balls after an elastic collision can be calculated using the conservation of momentum and kinetic energy, which remains constant in such a collision. The new direction of motion for the balls will depend on the sign of their final velocities. However, without further information, the time of collision cannot be determined.
Step-by-step explanation:
When dealing with a perfectly elastic collision, two important physical principles need to be applied: the conservation of momentum and the conservation of kinetic energy. Since no external forces are acting on the system of the two balls during the collision, we can use these principles to solve for the final velocities.
Final Velocities Calculation
Let m1 be the mass of the moving ball (0.586 kg), v1 be its initial velocity (3.90 m/s), m2 be the mass of the ball at rest (0.293 kg), and v2 be its initial velocity (0 m/s). After the collision, the velocities change to v1' and v2' for balls 1 and 2, respectively.
Using conservation of momentum and kinetic energy:
m1*v1 + m2*v2 = m1*v1' + m2*v2'
0.5*m1*v1^2 + 0.5*m2*v2^2 = 0.5*m1*v1'^2 + 0.5*m2*v2'^2
Solving these equations simultaneously, considering ball 2 was at rest:
v1' = (m1 - m2) / (m1 + m2) * v1
v2' = (2*m1) / (m1 + m2) * v1
Total Kinetic Energy After Collision
The total kinetic energy after the collision is the sum of the kinetic energies of both balls:
KE_total' = 0.5*m1*v1'^2 + 0.5*m2*v2'^2
Since the collision is elastic, KE_total' equals the initial kinetic energy of the system, which is solely the kinetic energy of ball 1 before the collision.
Direction of Motion After Collision
The new direction of motion for both balls will be determined by their final velocities. If the final velocity is positive, the ball moves east, and if negative, it moves west.
Unfortunately, without details about the coefficient of restitution or additional specifics, the time of collision cannot be determined from the information provided.