Blocks A A (mass 6.50kg kg ) and B B (mass 10.50kg kg ) move on a frictionless, horizontal surface. Initially, block B B is at rest and block A A is moving toward it at 4.00m/s m/s . The blocks are equipped with ideal spring bumpers. The collision is head-on, so all motion before and after the collision is along a straight line. Let +x +x be the direction of the initial motion of A A. Find the velocity of each block after they have moved apart.

Find the velocity of A A.

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To find the velocities of blocks A and B after they have moved apart, we can use the principle of conservation of momentum.

The momentum of an object is defined as the product of its mass and velocity. In this case, the initial momentum of block A is given by:
P_initial_A = mass_A * velocity_A

And the initial momentum of block B is given by:
P_initial_B = mass_B * velocity_B

Since there is no external force acting on the system, the total momentum before the collision is equal to the total momentum after the collision. Therefore, we have:
P_initial_A + P_initial_B = P_final_A + P_final_B

Now, let's substitute the given values into the equation. The masses of the blocks are given as:
mass_A = 6.50 kg
mass_B = 10.50 kg

The initial velocity of block A is given as 4.00 m/s. Since block B is initially at rest (velocity_B = 0), we can rewrite the equation as:
mass_A * velocity_A + mass_B * 0 = mass_A * velocity_final_A + mass_B * velocity_final_B

Simplifying the equation, we can eliminate the term mass_B * 0 and rewrite it as:
mass_A * velocity_A = mass_A * velocity_final_A + mass_B * velocity_final_B

Now, we need one more equation to solve for the velocities of blocks A and B. This is where the property of springs comes into play. Since the collision is head-on and the blocks are equipped with ideal spring bumpers, we can assume that the collision is an elastic collision, meaning that both momentum and kinetic energy are conserved.

In an elastic collision, we also have the relationship:
(1/2) * mass_A * (velocity_A)^2 + (1/2) * mass_B * (velocity_B)^2 = (1/2) * mass_A * (velocity_final_A)^2 + (1/2) * mass_B * (velocity_final_B)^2

Again, we can substitute the given values and simplify the equation to solve for the velocities of blocks A and B.

By solving these two equations simultaneously, you can find the velocities of blocks A and B after they have moved apart.