In Figure 9-61, block 1 (mass 2.0 kg) is moving rightward at 8 m/s and block 2 (mass 4.8 kg) is moving rightward at 2.8 m/s. The surface is frictionless, and a spring with spring constant of 1180 N/m is fixed to block 2. When the blocks collide, the compression of the spring is maximum at the instant the blocks have the same velocity. Find the maximum compression.

To find the maximum compression of the spring, we need to apply the principle of conservation of momentum and the principle of conservation of mechanical energy.

First, let's calculate the initial momentum of the system. The momentum is given by the product of mass and velocity:

Initial momentum = mass × velocity

The initial momentum of block 1 (m1) is given by:
momentum1 = m1 × v1

momentum1 = 2.0 kg × 8 m/s
momentum1 = 16.0 kg⋅m/s

Similarly, the initial momentum of block 2 (m2) is given by:
momentum2 = m2 × v2

momentum2 = 4.8 kg × 2.8 m/s
momentum2 = 13.44 kg⋅m/s

The total initial momentum of the system is the sum of the initial momentum of block 1 and block 2:
total initial momentum = momentum1 + momentum2

total initial momentum = 16.0 kg⋅m/s + 13.44 kg⋅m/s
total initial momentum = 29.44 kg⋅m/s

At the instant the blocks have the same velocity, the principle of conservation of momentum tells us that the final momentum of the system will also be equal to 29.44 kg⋅m/s (since no external forces are acting on the system).

Now, let's calculate the maximum compression of the spring using the principle of conservation of mechanical energy.

The mechanical energy of the system consists of the kinetic energy of the blocks and the potential energy stored in the spring.

Initially, the kinetic energy of the system is given by:
initial kinetic energy = 1/2 × m1 × v1^2 + 1/2 × m2 × v2^2

initial kinetic energy = 1/2 × 2.0 kg × (8 m/s)^2 + 1/2 × 4.8 kg × (2.8 m/s)^2
initial kinetic energy = 64 J + 18.816 J
initial kinetic energy = 82.816 J

At the maximum compression of the spring, all the initial kinetic energy will be converted into potential energy stored in the spring. So, the final potential energy of the system is equal to the initial kinetic energy.

Finally, let's calculate the maximum compression of the spring using the formula for potential energy stored in a spring:

Potential energy = 1/2 × k × x^2

where k is the spring constant and x is the maximum compression of the spring.

Equating the initial kinetic energy to the potential energy of the spring:
82.816 J = 1/2 × 1180 N/m × x^2

Simplifying the equation:
x^2 = 2 × 82.816 J / 1180 N/m
x^2 = 140.2736 J/m / 1180 N/m
x^2 ≈ 0.1189 m^2

Taking the square root of both sides to find x:
x ≈ √(0.1189 m^2)
x ≈ 0.344 m

Therefore, the maximum compression of the spring is approximately 0.344 meters.

To find the maximum compression of the spring, we need to equate the kinetic energy of the blocks before and after the collision.

Given:
Mass of block 1 (m1) = 2.0 kg
Initial velocity of block 1 (v1i) = 8 m/s
Mass of block 2 (m2) = 4.8 kg
Initial velocity of block 2 (v2i) = 2.8 m/s
Spring constant (k) = 1180 N/m

Step 1: Calculate the initial total kinetic energy of the system:
Initial kinetic energy (KEi) = (1/2) * m1 * v1i^2 + (1/2) * m2 * v2i^2

KEi = (1/2) * 2.0 * (8^2) + (1/2) * 4.8 * (2.8^2)
KEi = 64 + 18.816
KEi = 82.816 J

Step 2: Calculate the final velocity of the blocks after the collision.
Since the blocks have the same velocity at maximum compression, we can calculate it using conservation of momentum:
m1 * v1f + m2 * v2f = 0

2.0 * v1f + 4.8 * v2f = 0

Step 3: Use the concept of conservation of kinetic energy to find the final kinetic energy of the system.
Final kinetic energy (KEf) = (1/2) * m1 * v1f^2 + (1/2) * m2 * v2f^2

Step 4: Calculate the potential energy stored in the spring at maximum compression.
Potential energy (PE) = (1/2) * k * x^2 [where x is the maximum compression]

Step 5: Equate the initial kinetic energy (KEi) to the sum of the final kinetic energy (KEf) and the potential energy (PE).
KEi = KEf + PE

82.816 = (1/2) * m1 * v1f^2 + (1/2) * m2 * v2f^2 + (1/2) * k * x^2

Step 6: Solve for x, the maximum compression of the spring.
Rearranging the equation, we have:
x^2 = (2 * KEi - (m1 * v1f^2 + m2 * v2f^2)) / k

x^2 = (2 * 82.816 - (2.0 * v1f^2 + 4.8 * v2f^2)) / (1180)

Step 7: Substitute the value of v1f from the conservation of momentum equation.
2.0 * v1f + 4.8 * v2f = 0

v1f = -(4.8 * v2f) / 2.0

Step 8: Substitute the value of v1f in the equation for x^2.
x^2 = (2 * 82.816 - (2.0 * (-(4.8 * v2f) / 2.0)^2 + 4.8 * v2f^2)) / (1180)

Simplify the expression:
x^2 = (2 * 82.816 - (2.0 * (23.04 * v2f^2) + 4.8 * v2f^2)) / (1180)

Step 9: Solve the equation to find the maximum compression.
Solve the equation for x to find the maximum compression of the spring. The equation may not have a closed-form solution, so you may need to use numerical methods or a calculator to find the value of x.

Using conservation of momentum:

2*8 + 4.8*2.8 = (2+4.8)*vf = 6.8*vf

Solve for vf, the final speed of the blocks when they have the same velocity.

1/2*k*x^2 = 1/2*m*vf^2

Solve for x; k is the spring constant = 1180 N/m; x is the maximum compression of the spring