A bullet of mass m=0.05kg embeds itself in a block of mass M=1.2kg, which attached to a spring of force constant k=245N/m. If the initial speed of the bullet v0=1.32m/s, find the maximum compression of the spring

the momentum of the bullet becomes the momentum of the block/bullet

... 0.05 * 1.32 = Vb * (1.2 + 0.05)

find the initial energy (kinetic) of the block/bullet
... this is the energy that compresses the spring

1/2 m v^2 = 1/2 k x^2

To find the maximum compression of the spring, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of the system remains constant throughout the process, neglecting the effects of external forces such as friction.

Initially, the only forms of energy are the kinetic energy of the bullet and the potential energy stored in the spring. Since the block is at rest, it has no initial kinetic energy.

The formula for kinetic energy is given by:
KE = 1/2 * m * v^2

The formula for potential energy stored in a spring is given by:
PE = 1/2 * k * x^2

Where:
- KE is the kinetic energy
- m is the mass
- v is the velocity
- PE is the potential energy
- k is the force constant of the spring
- x is the displacement or compression of the spring

At the maximum compression of the spring, all the initial kinetic energy of the bullet is converted into potential energy stored in the spring.

Using the principle of conservation of mechanical energy, we can equate the initial kinetic energy of the bullet to the potential energy stored in the spring at maximum compression:

1/2 * m * v0^2 = 1/2 * k * x^2

Plugging in the given values:
1/2 * 0.05kg * (1.32m/s)^2 = 1/2 * 245N/m * x^2

Simplifying the equation, we get:

0.033kg * (1.74m^2/s^2) = 0.1225N/m * x^2

Multiplying the numbers:

0.05742 kg⋅m^2/s^2 = 0.1225N/m * x^2

Dividing both sides by 0.1225 N/m:

0.4697 kg⋅m^2/s^2 = x^2

Taking the square root of both sides:

x = sqrt(0.4697 kg⋅m^2/s^2)

Evaluating the square root:

x ≈ 0.686 m

Therefore, the maximum compression of the spring is approximately 0.686 meters.