how is this equation derived

V'2= V1(2M1)/(M1+M2) (elastic collision - where one object is moving and the other is stationary)

It is derived by requiring that both momentum and kinetic energy be conserved.

The center of mass moves forward at velocity vcm = M1*V1/(M1 + M2). After collision, the CM retains that velocity but the velocities change directions in CM-fixed coordinates.

In CM-fixed coordinates,
v2 = -V1*M1/(M1+M2)
v2' (after collision) = V1*M1/(M1+M2)

In lab-fixed coordinates
V2' = v2 -vcm
= 2 V1*M1/(M1+M2)

To derive the equation for the final velocity (V'2) after an elastic collision between an object with initial velocity (V1) and mass (M1) with an object at rest with mass (M2), we can apply the principles of conservation of momentum and conservation of kinetic energy.

1. Conservation of momentum:
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be represented as:
Momentum before collision = Momentum after collision

The momentum of an object with mass (m) and velocity (v) is given by the equation: p = mv.

Therefore, the momentum before the collision (p1) is given by:
p1 = M1 * V1

Since the second object is initially at rest, its momentum before the collision (p2) is zero:
p2 = 0

After the collision, the momentum of the first object (p'1) and the second object (p'2) can be given as:
p'1 = M1 * V'1
p'2 = M2 * V'2

According to the conservation of momentum, we have:
p1 + p2 = p'1 + p'2
M1 * V1 + 0 = M1 * V'1 + M2 * V'2
M1 * V1 = M1 * V'1 + M2 * V'2

2. Conservation of kinetic energy:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Mathematically, this can be represented as:
Kinetic energy before collision = Kinetic energy after collision

The kinetic energy of an object with mass (m) and velocity (v) can be given by the equation: KE = (1/2) * m * v^2.

Therefore, the kinetic energy before the collision (KE1) is given by:
KE1 = (1/2) * M1 * V1^2

After the collision, the kinetic energy of the first object (KE'1) and the second object (KE'2) can be given as:
KE'1 = (1/2) * M1 * V'1^2
KE'2 = (1/2) * M2 * V'2^2

According to the conservation of kinetic energy, we have:
KE1 = KE'1 + KE'2
(1/2) * M1 * V1^2 = (1/2) * M1 * V'1^2 + (1/2) * M2 * V'2^2
M1 * V1^2 = M1 * V'1^2 + M2 * V'2^2

Now, we have two equations from the conservation of momentum and the conservation of kinetic energy. We can solve this system of equations to find the expression for V'2.

Using the equation: M1 * V1 = M1 * V'1 + M2 * V'2, we can rearrange it as:
M1 * V'1 = M1 * V1 - M2 * V'2

Substitute this expression for M1 * V'1 in the equation M1 * V1^2 = M1 * V'1^2 + M2 * V'2^2:
M1 * V1^2 = (M1 * V1 - M2 * V'2)^2 + M2 * V'2^2

Expanding and simplifying the equation, we get:
M1 * V1^2 = M1^2 * V1^2 - 2 * M1 * M2 * V1 * V'2 + M2^2 * V'2^2 + M2 * V'2^2

Collecting like terms and simplifying further, we have:
0 = M1^2 * V1^2 - 2 * M1 * M2 * V1 * V'2 + (M2^2 + M2) * V'2^2

Rearranging the equation, we get:
2 * M1 * M2 * V1 * V'2 = (M1^2 + M2^2 + M2) * V'2^2 - M1^2 * V1^2

Finally, solving for V'2, we arrive at the equation:
V'2 = V1 * (2 * M1) / (M1 + M2)

Hence, we have derived the equation for the final velocity (V'2) in an elastic collision between an object with initial velocity (V1) and mass (M1) with an object at rest with mass (M2).