A 4.00-g bullet is moving horizontally with a velocity vector v = 348 m/s, where the sign + indicates that it is moving to the right (see part a of the drawing). The bullet is approaching two blocks resting on a horizontal frictionless surface. Air resistance is negligible. The bullet passes completely through the first block (an inelastic collision) and embeds itself in the second one, as indicated in part b. Note that both blocks are moving after the collision with the bullet. The mass of the first block is 1150 g, and its velocity is +0.594 m/s after the bullet passes through it. The mass of the second block is 1530 g. (a) What is the velocity of the second block after the bullet imbeds itself? (b) Find the ratio of the total kinetic energy after the collision to that before the collision.

To answer this question, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.

(a) To find the velocity of the second block after the bullet embeds itself, we can use the principle of conservation of momentum.

The principle of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. In this case, the momentum before the collision includes the momentum of the bullet and the first block, and after the collision, it includes the momentum of both blocks.

The momentum of an object is given by the product of its mass and velocity: momentum = mass * velocity.

Before the collision:
Total momentum before = momentum of bullet + momentum of first block
Total momentum before = (mass of bullet * velocity) + (mass of first block * velocity)

After the collision:
Total momentum after = momentum of first block + momentum of second block
Total momentum after = (mass of first block * final velocity of first block) + (mass of second block * final velocity of second block)

Since the bullet embeds itself in the second block, the final velocity of the second block will include the velocity of the bullet.

Equating the total momentum before and after the collision, we have:

(mass of bullet * velocity) + (mass of first block * velocity) = (mass of first block * final velocity of first block) + (mass of second block * final velocity of second block)

Plug in the given values:

(4.00 g * 348 m/s) + (1150 g * 0.594 m/s) = (1150 g * 0.594 m/s) + (1530 g * final velocity of second block)

Solving this equation will give us the velocity of the second block after the bullet embeds itself.

(b) To find the ratio of the total kinetic energy after the collision to that before the collision, we can use the principle of conservation of kinetic energy.

The principle of conservation of kinetic energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision. In this case, the kinetic energy before the collision includes the kinetic energy of the bullet and the first block, and after the collision, it includes the kinetic energy of both blocks.

The kinetic energy of an object is given by the equation: kinetic energy = (1/2) * mass * velocity^2.

Before the collision:
Total kinetic energy before = kinetic energy of bullet + kinetic energy of first block
Total kinetic energy before = (1/2) * mass of bullet * velocity^2 + (1/2) * mass of first block * velocity^2

After the collision:
Total kinetic energy after = kinetic energy of first block + kinetic energy of second block
Total kinetic energy after = (1/2) * mass of first block * final velocity of first block^2 + (1/2) * mass of second block * final velocity of second block^2

We can calculate the total kinetic energy before and after the collision using the given values, and then find the ratio of the total kinetic energy after to that before.

So, we can solve both parts of the question by applying the principles of conservation of momentum and conservation of kinetic energy.