Two objects with masses 2.0 and 3.0 kg move toward each other, both with speeds V0 = 5.0 m/s. They collide head on and stick together.

(a) Calculate their final velocity.
(b) Calculate the amount of kinetic energy lost during the process.
(c) Suppose the two masses approach each other at 90° before the collision (e.g., along the x- and y-axes). What will be the kinetic energy loss in this case?

To solve these questions, we can use the principles of conservation of momentum and conservation of kinetic energy.

(a) To calculate the final velocity of the objects after they collide and stick together, we can use the conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

The momentum of an object is given by the product of its mass and velocity. So, before the collision, the total momentum is:

Momentum before = (mass1 * velocity1) + (mass2 * velocity2)

= (2.0 kg * 5.0 m/s) + (3.0 kg * -5.0 m/s) [As the objects are moving towards each other, we take opposite signs for velocities]

= 10 kg m/s - 15 kg m/s

= -5 kg m/s

After the collision, the two masses stick together, so they move with the same final velocity, which we'll call Vf. Therefore, the total momentum after the collision is:

Momentum after = (mass1 + mass2) * Vf [as the masses stick together]

= (2.0 kg + 3.0 kg) * Vf

= 5.0 kg * Vf

According to the conservation of momentum, the momentum before and after the collision should be equal:

-5 kg m/s = 5.0 kg * Vf

Solving for Vf, we get:

Vf = -1 m/s

Therefore, the final velocity of the two objects after the collision is -1 m/s.

(b) To calculate the amount of kinetic energy lost during the process, we can use the conservation of kinetic energy principle. The formula for kinetic energy is given by:

Kinetic energy = 0.5 * mass * (velocity^2)

Before the collision, the total kinetic energy is the sum of the kinetic energies of both objects:

Kinetic energy before = (0.5 * 2.0 kg * (5.0 m/s)^2) + (0.5 * 3.0 kg * (-5.0 m/s)^2)

= 25.0 J + 37.5 J

= 62.5 J

After the collision, the two masses stick together, so they have a final velocity of -1 m/s. Therefore, the total kinetic energy after the collision is:

Kinetic energy after = 0.5 * (2.0 kg + 3.0 kg) * (-1 m/s)^2

= 0.5 * 5.0 kg * 1.0 m/s^2

= 2.5 J

The amount of kinetic energy lost during the process is the difference between the initial and final kinetic energies:

Kinetic energy lost = Kinetic energy before - Kinetic energy after

= 62.5 J - 2.5 J

= 60 J

Therefore, the amount of kinetic energy lost during the process is 60 Joules.

(c) If the two masses approach each other at 90° before the collision, the collision would be an oblique collision. In this case, the kinetic energy loss may be different. To calculate the kinetic energy loss for an oblique collision, we need to know the angle and the masses involved. Since the question does not provide these details, we cannot provide a specific value for the kinetic energy loss in this scenario.

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