A 0.02kg bullet collides with a 5.75kg pendulum. After the collision, the pair swings up to a maximum height of 0.386m. Determine the velocity of the bullet just before impact.

To determine the velocity of the bullet just before impact, we can use the principles of conservation of momentum and conservation of mechanical energy.

First, let's define our variables:
m1 = mass of the bullet = 0.02 kg
m2 = mass of the pendulum = 5.75 kg
h = maximum height reached by the bullet-pendulum system after the collision = 0.386 m
v1 = velocity of the bullet just before impact (what we want to find)
v2 = velocity of the pendulum just before impact

Now let's set up the equations based on the conservation principles:

Conservation of momentum:
m1 * v1 + m2 * v2 = 0

Conservation of mechanical energy:
The initial mechanical energy before the collision is equal to the final mechanical energy at the maximum height reached:
(1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 = m1 * g * h

In these equations, g represents the acceleration due to gravity, which is approximately 9.8 m/s^2.

We have two equations with two unknowns (v1 and v2), so we can solve this system of equations simultaneously.

From the conservation of momentum equation:
v2 = - (m1 * v1) / m2

Substituting this into the conservation of mechanical energy equation:
(1/2) * m1 * v1^2 + (1/2) * m2 * (-(m1 * v1) / m2)^2 = m1 * g * h

Simplifying this equation gives:
(1/2) * m1 * v1^2 + (1/2) * (m1 * v1)^2 / m2 = m1 * g * h

Let's substitute the given values into the equation and solve for v1:

(1/2) * 0.02 * v1^2 + (1/2) * (0.02 * v1)^2 / 5.75 = 0.02 * 9.8 * 0.386

0.01 * v1^2 + 0.01 * v1^2 / 5.75 = 0.019604

Multiplying through by 5.75 gives:
0.0575 * v1^2 + 0.01 * v1^2 = 0.112545

Combining like terms:
0.0675 * v1^2 = 0.112545

Dividing by 0.0675 gives:
v1^2 = 1.665925925925926

Taking the square root of both sides gives:
v1 ≈ 1.29 m/s

Therefore, the velocity of the bullet just before impact is approximately 1.29 m/s.