I have read multiple threads including:

When is energy conserved in a collision and not momentum?

How to calculate velocities after collision?

How can I calculate the final velocities of two spheres after an elastic collision?

Calculating new velocities of $n$-dimensional particles after collision

Velocities in an elastic collision

Summation of the velocities before and after an elastic collision

In an elastic collision, I understand that momentum is conserved and kinetic energy is conserved. If billiard ball of silver (with velocity $v(Ag)$ impacts a stationary billiard ball of aluminum, I am trying to calculate the velocity of the aluminum ball after the collision, $v(Al)$. After an elastic collision, the impactor is at rest and the impactee has the motion.

Using momentum, $= m \cdot v$

$$m(Ag) \cdot v(Ag) = m(Al) \cdot v(Al)$$

Assuming silver is 4x denser than aluminum, then using momentum, the aluminum ball should have velocity

$$v(Al) = 4\cdot v(Ag)$$

But if we use kinetic energy, $1/2 m \cdot v^2$

$$\frac12m(Ag)\cdot v(ag)^2=\frac12m(Al)\cdot v(Al)^2$$

$$v(Al)^2=\frac{m(Ag)}{m(Al)}\cdot v(Ag)^2$$

$$v(Al)=\left(\frac{m(Ag)}{m(Al)}\right)^{\frac12}\cdot v(Ag)$$

$$v(Al)=2\cdot v(Ag)$$

Somewhere I have lost some neuron connections in my brain because I cannot resolve this conflict. This is a perfectly elastic collision so both momentum and kinetic energy should be conserved.

In an elastic collision, I understand that momentum is conserved and kinetic energy is conserved. If billiard ball of silver (with velocity $v_{(Ag)}$ impacts a stationary billiard ball of aluminum, I am trying to calculate the velocity of the aluminum ball after the collision, $v_{(Al)}$. After an elastic collision, the impactor is at rest and the impactee has the motion.

Using momentum, $= m \cdot v$

$$m_{(Ag)} \cdot v_{(Ag)} = m_{(Al)} \cdot v_{(Al)}$$

Assuming silver is 4x denser than aluminium, then using momentum, the aluminium ball should have velocity

$$v_{(Al)} = 4\cdot v_{(Ag)}$$

But if we use kinetic energy, $1/2 m \cdot v^2$

$$\frac12m_{(Ag)}\cdot v_{(ag)}^2=\frac12m_{(Al)}\cdot v_{(Al)}^2$$

$$v_{(Al)}^2=\frac{m_{(Ag)}}{m_{(Al)}}\cdot v_{(Ag)}^2$$

$$v_{(Al)}=\left(\frac{m_{(Ag)}}{m_{(Al)}}\right)^{\frac12}\cdot v_{(Ag)}$$

$$v_{(Al)}=2\cdot v_{(Ag)}$$

Somewhere I have lost some neuron connections in my brain because I cannot resolve this conflict. This is a perfectly elastic collision so both momentum and kinetic energy should be conserved.

I have read multiple threads including:

When is energy conserved in a collision and not momentum?

How to calculate velocities after collision?

How can I calculate the final velocities of two spheres after an elastic collision?

Calculating new velocities of $n$-dimensional particles after collision

Velocities in an elastic collision

Summation of the velocities before and after an elastic collision

I have read multiple threads including:

When is energy conserved in a collision and not momentum?

How to calculate velocities after collision?

How can I calculate the final velocities of two spheres after an elastic collision?

Calculating new velocities of $n$-dimensional particles after collision

Velocities in an elastic collision

Summation of the velocities before and after an elastic collision

In an elastic collision, I understand that momentum is conserved and kinetic energy is conserved. If billiard ball of silver( with velocity v(Ag) impacts a stationary billiard ball of aluminum, I am trying to calculate the velocity of the aluminum ball after the collision, v(Al). After an elastic collision  , the impactor is at rest and the impactee has the motion.

Using momentum, = m * v

m(Ag) * v(Ag) = m(Al) * v(Al).

Assuming silver is 4x denser than aluminum, then using momentum, the aluminum ball should have velocity

v(Al) = 4 v(Ag).

But if we use kinetic energy, 1/2 m * v^2

1/2 m(Ag) * v(ag) ^ 2 = 1/2 m(Al) * v(Al) ^ 2

v(Al) ^ 2 = m(Ag)/m(Al) * v(Ag) ^ 2

v(Al) = ( m(Ag)/m(Al) ) ^ 1/2 * v(Ag)

v(Al) = 2 * v(Ag)

Somewhere I have lost some neuron connections in my brain because I cannot resolve this conflict. This is a perfectly elastic collision so both momentum and kinetic energy should be conserved.

I have read multiple threads including:

When is energy conserved in a collision and not momentum?

How to calculate velocities after collision?

How can I calculate the final velocities of two spheres after an elastic collision?

Calculating new velocities of $n$-dimensional particles after collision

Velocities in an elastic collision

Summation of the velocities before and after an elastic collision

In an elastic collision, I understand that momentum is conserved and kinetic energy is conserved. If billiard ball of silver  (with velocity $v(Ag)$ impacts a stationary billiard ball of aluminum, I am trying to calculate the velocity of the aluminum ball after the collision, $v(Al)$. After an elastic collision, the impactor is at rest and the impactee has the motion.

Using momentum, $= m \cdot v$

$$m(Ag) \cdot v(Ag) = m(Al) \cdot v(Al)$$

Assuming silver is 4x denser than aluminum, then using momentum, the aluminum ball should have velocity

$$v(Al) = 4\cdot v(Ag)$$

But if we use kinetic energy, $1/2 m \cdot v^2$

$$\frac12m(Ag)\cdot v(ag)^2=\frac12m(Al)\cdot v(Al)^2$$

$$v(Al)^2=\frac{m(Ag)}{m(Al)}\cdot v(Ag)^2$$

$$v(Al)=\left(\frac{m(Ag)}{m(Al)}\right)^{\frac12}\cdot v(Ag)$$

$$v(Al)=2\cdot v(Ag)$$

Somewhere I have lost some neuron connections in my brain because I cannot resolve this conflict. This is a perfectly elastic collision so both momentum and kinetic energy should be conserved.

I have read multiple threads including:

When is energy conserved in a collision and not momentum?

How to calculate velocities after collision?

How can I calculate the final velocities of two spheres after an elastic collision?

Calculating new velocities of $n$-dimensional particles after collision

Velocities in an elastic collision

Summation of the velocities before and after an elastic collision

In an elastic collision, I understand that momentum is conserved and kinetic energy is conserved. If billiard ball of silver( with velocity v(Ag) impacts a stationary billiard ball of aluminum, I am trying to calculate the velocity of the aluminum ball after the collision, v(Al). After an elastic collision , the impactor is at rest and the impactee has the motion.

Using momentum, = m * v

m(Ag) * v(Ag) = m(Al) * v(Al).

Assuming silver is 4x denser than aluminum, then using momentum, the aluminum ball should have velocity

v(Al) = 4 v(Ag).

But if we use kinetic energy, 1/2 m * v^2

1/2 m(Ag) * v(ag) ^ 2 = 1/2 m(Al) * v(Al) ^ 2

v(Al) ^ 2 = m(Ag)/m(Al) * v(Ag) ^ 2

v(Al) = ( m(Ag)/m(Al) ) ^ 1/2 * v(Ag)

v(Al) = 2 * v(Ag)

Somewhere I have lost some neuron connections in my brain because I cannot resolve this conflict. This is a perfectly elastic collision so both momentum and kinetic energy should be conserved.

I have read multiple threads including:

When is energy conserved in a collision and not momentum?

How to calculate velocities after collision?

How can I calculate the final velocities of two spheres after an elastic collision?

Calculating new velocities of $n$-dimensional particles after collision

Velocities in an elastic collision

Summation of the velocities before and after an elastic collision

In an elastic collision, I understand that momentum is conserved and kinetic energy is conserved. If billiard ball of silver( with velocity v(Ag) impacts a stationary billiard ball of aluminum, I am trying to calculate the velocity of the aluminum ball after the collision, v(Al). After an elastic collision , the impactor is at rest and the impactee has the motion.

Using momentum, = m * v

m(Ag) * v(Ag) = m(Al) * v(Al).

Assuming silver is 4x denser than aluminum, then using momentum, the aluminum ball should have velocity

v(Al) = 4 v(Ag).

But if we use kinetic energy, 1/2 m * v^2

1/2 m(Ag) * v(ag) ^ 2 = 1/2 m(Al) * v(Al) ^ 2

v(Al) ^ 2 = m(Ag)/m(Al) * v(Ag) ^ 2

v(Al) = ( m(Ag)/m(Al) ) ^ 1/2 * v(Ag)

v(Al) = 2 * v(Ag)

Somewhere I have lost some neuron connections in my brain because I cannot resolve this conflict. This is a perfectly elastic collision so both momentum and kinetic energy should be conserved.