A 1.0kg object traveling at 1.0m/s collides head on with a 2.0kg object initially at rest . You are required to find the velocity of each object after impact if the collision is perfectly elastic?

In a perfectly elastic collision, both momentum and kinetic energy are conserved.

Let the velocity of the 1.0kg object after the collision be v1 and the velocity of the 2.0kg object after the collision be v2.

Using the conservation of momentum:
Initial momentum = Final momentum
(1.0kg)(1.0m/s) = (1.0kg)(v1) + (2.0kg)(v2)

1.0 = v1 + 2v2

Using the conservation of kinetic energy:
Initial kinetic energy = Final kinetic energy
(1/2)(1.0kg)(1.0m/s)^2 = (1/2)(1.0kg)(v1)^2 + (1/2)(2.0kg)(v2)^2

0.5 = 0.5v1^2 + v2^2

Now we have two equations:
1 = v1 + 2v2
0.5 = 0.5v1^2 + v2^2

Solving these equations simultaneously, we get:
v1 = 1.5 m/s
v2 = -0.25 m/s

Therefore, the velocity of the 1.0kg object after the collision is 1.5 m/s and the velocity of the 2.0kg object after the collision is -0.25 m/s.

Step 1: Momentum Conservation

In a perfectly elastic collision, the total momentum of the system is conserved before and after the collision:

```
P_before = P_after
```

where P is the momentum (mass x velocity).

Step 2: Initial Momentim

Before the collision, the momentum of the 1.0kg object is:

```
P_1_before = 1.0 kg x 1.0 m/s = 1.0 kg m/s
```

The initial momentum of the 2.0kg object is 0 since it's initially at rest.

Step 3: Solve for After-Collision Velocities

Let v_1' and v_2' be the velocities of the 1.0kg and 2.0kg objects after the collision, respectively. Then, the total momentum after the collision is:

```
P_after = P_1_after + P_2_after = 1.0 kg m/s + 2.0 kg x v_2'
```

Equating this to P_before, we get:

```
1.0 kg m/s = 1.0 kg x v_1' + 2.0 kg x v_2'
```

Step 4: Energy Conservation

In a perfectly elastic collision, the total kinetic energy is also conserved:

```
K_before = K_after
```

where K is the kinetic energy (1/2 - mass - velocity^2).

Step 5: Solve for Velocities

Solving the conservation of momentum and energy equations simultaneously, we get:

```
v_1' = 0.667 m/s
v_2' = 0.333 m/s
```

Therefore, the velocity of the 1.0kg object after the collision is 0.667 m/s, and the velocity of the 2.0kg object after the collision is 0.333 m/s.