Consider a frictionless track as shown in the figure below. A block of mass m1 = 4.20 kg is released from A. It makes a head on elastic collision at B with a block of mass m2 = 9.00 kg that is initially at rest. Calculate the maximum height to which m1 rises after the collision.

To calculate the maximum height to which m1 rises after the collision, we can use the principle of conservation of mechanical energy.

Step 1: Calculate the initial velocity of block m1.
Since block m1 is released from rest, its initial velocity is zero.

Step 2: Calculate the initial velocity of block m2.
Since block m2 is initially at rest, its initial velocity is also zero.

Step 3: Determine the velocities of the blocks after the collision using the principle of conservation of momentum.
The conservation of momentum equation can be written as:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f

Since block m1 is moving to the left and block m2 is initially at rest, we can rewrite the equation as:
m1 * v1i = m1 * v1f + m2 * v2f

Step 4: Apply the conservation of kinetic energy equation after the collision.
Since the collision is elastic, the total kinetic energy of the system is conserved.
The equation for conservation of kinetic energy is:
(1/2) * m1 * v1f^2 + (1/2) * m2 * v2f^2 = (1/2) * m1 * v1i^2 + (1/2) * m2 * v2i^2

Step 5: Apply the conservation of mechanical energy equation after the collision.
Since the track is frictionless, the system's mechanical energy is conserved and converted between kinetic and potential energies.
The equation for conservation of mechanical energy is:
m1 * g * h = (1/2) * m1 * v1f^2 + (1/2) * m2 * v2f^2

Step 6: Solve the conservation of momentum equation for v1f.
m1 * v1i = m1 * v1f + m2 * v2f
v1f = (m1 * v1i - m2 * v2f) / m1

Step 7: Substitute the value of v1f into the equation for conservation of mechanical energy.
m1 * g * h = (1/2) * m1 * ((m1 * v1i - m2 * v2f) / m1)^2 + (1/2) * m2 * v2f^2

Step 8: Simplify the equation and solve for h.
m1 * g * h = (1/2) * m1 * (v1i^2 - 2 * m1 * v1i * v2f + m2^2 * v2f^2) + (1/2) * m2 * v2f^2

Distribute and simplify the equation further:
m1 * g * h = (1/2) * m1 * v1i^2 - m1^2 * v1i * v2f + (1/2) * m1 * m2^2 * v2f^2 + (1/2) * m2 * v2f^2

Combine like terms:
m1 * g * h = (1/2) * m1 * v1i^2 + (1/2) * m2 * v2f^2 - m1^2 * v1i * v2f + (1/2) * m1 * m2^2 * v2f^2

Factor out v2f:
m1 * g * h = (1/2) * (m1 * v1i^2 + m2 * v2f^2) - (m1^2 * v1i - m1 * m2^2) * v2f

Simplify further:
m1 * g * h = (1/2) * (m1 * v1i^2 + m2 * v2f^2) - v2f * (m1^2 * v1i - m1 * m2^2)

Step 9: Solve for h.
h = [(1/2) * (m1 * v1i^2 + m2 * v2f^2) - v2f * (m1^2 * v1i - m1 * m2^2)] / (m1 * g)

Step 10: Calculate the values of m1, m2, v1i, and v2f from the given information and substitute them into the equation to find h.

To calculate the maximum height to which m1 rises after the collision, we can use the principle of conservation of mechanical energy.

1. First, we need to find the initial velocity of m1 just before the collision.
Since m1 is released from rest at point A, its initial velocity is 0 m/s.

2. Next, we can calculate the velocity of m1 just after the collision using the principle of conservation of linear momentum.
The equation for the conservation of momentum in the x-direction is:

m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f

Where:
m1, m2 = masses of the blocks
v1i, v1f = initial and final velocities of block m1
v2i, v2f = initial and final velocities of block m2

Since block m2 is at rest before the collision, v2i = 0.
Also, since it is mentioned that the collision is elastic, total mechanical energy is conserved. Therefore, the kinetic energy before the collision is equal to the kinetic energy after the collision.

3. Calculate the velocity of m2 just after the collision.
Since m2 is initially at rest and the collision is elastic, the velocity of m2 after the collision, v2f, can be calculated using the equation:

v2f = (m1 - m2) / (m1 + m2) * v1i

4. Calculate the velocity of m1 just after the collision.
Using the conservation of momentum equation mentioned earlier, we can find v1f:

v1f = (2 * m2 * v2i + v1i * (m1 - m2)) / (m1 + m2)

5. Calculate the maximum height reached by m1.
To find the maximum height reached by m1, we need to consider the conservation of mechanical energy. The potential energy at the maximum height is equal to the initial potential energy plus the change in kinetic energy.

The initial potential energy is zero since m1 is at the same height initially:
PE_initial = 0

The final potential energy at the maximum height is:
PE_final = m1 * g * h

And the change in kinetic energy is:
ΔKE = 0.5 * m1 * v1f^2

Since the total mechanical energy is conserved, we have:
PE_initial + ΔKE = PE_final

Solving for h (the maximum height):
h = (v1f^2) / (2 * g)

Where:
g = acceleration due to gravity (approximately 9.8 m/s^2)

By plugging in the values for the known quantities, you can calculate the maximum height to which m1 rises after the collision.