A 0.028 kg rifle bullet traveling 210 m/s buries itself in a 3.1 kg pendulum hanging on a 2.5 m long string, which makes the pendulum swing upward in an arc.

1. Determine the vertical and horizontal components of the pendulum's displacement.

To determine the vertical and horizontal components of the pendulum's displacement, we need to analyze the situation and break down the motion into its components.

First, let's consider the impact of the bullet on the pendulum. The bullet has a mass of 0.028 kg and is traveling at a speed of 210 m/s. Upon impact, it buries itself in the pendulum, so we can assume that its mass is added to the pendulum's mass.

The total mass of the pendulum and the bullet after the impact is the sum of their masses, which is:
Total mass = mass of pendulum + mass of bullet
Total mass = 3.1 kg + 0.028 kg
Total mass = 3.128 kg

Now, let's analyze the motion of the pendulum after the impact. Since the pendulum swings upward in an arc, we can decompose its displacement into vertical and horizontal components.

The vertical component of the pendulum's displacement is the distance it moves vertically upward. This vertical displacement can be calculated using the conservation of energy principle. The bullet's initial kinetic energy is converted into potential energy in the pendulum.

The potential energy gained by the pendulum is given by:
Potential energy = mass * gravity * height

Since the pendulum swings in an arc, the height is given by the length of the string (2.5 m). The gravitational acceleration is approximately 9.8 m/s².

Now, substituting the values we know into the formula, we can determine the vertical displacement:
Vertical displacement = (Total mass * gravity * height) / Total mass
Vertical displacement = (3.128 kg * 9.8 m/s² * 2.5 m) / 3.128 kg

Simplifying, we find that the vertical displacement is equal to:
Vertical displacement ≈ 19.49 m

The horizontal component of the pendulum's displacement is the distance it moves horizontally. Since the pendulum swings in an arc, the horizontal displacement is equal to the initial horizontal velocity of the bullet multiplied by the time of impact.

To find the time of impact, we can use the equation of motion:
Final velocity = Initial velocity + acceleration * time

In this case, the final velocity is zero because the bullet is buried inside the pendulum. The initial velocity is given as 210 m/s, and the acceleration can be calculated as the force acting on the bullet divided by the total mass.

F = m * a (Newton's second law)

The force acting on the bullet can be calculated as the change in momentum:
Force = mass of bullet * change in velocity
Change in velocity = final velocity - initial velocity (since the bullet comes to rest after impact)

Now, substituting the values we know into the equations, we can determine the change in velocity and the force:
Change in velocity = 0 m/s - 210 m/s
Change in velocity = -210 m/s
Force = (0.028 kg * -210 m/s)

Now, we can calculate the acceleration:
F = m * a
(0.028 kg * -210 m/s) = 3.128 kg * a

Simplifying, we find that the acceleration is equal to:
Acceleration ≈ -0.572 m/s²

Now, we can use the equation of motion to find the time of impact:
0 = 210 m/s + (-0.572 m/s²) * time

Solving for time, we find:
Time ≈ 366.433 s

Finally, we can determine the horizontal displacement:
Horizontal displacement = initial horizontal velocity * time of impact
Horizontal displacement = 210 m/s * 366.433 s

Simplifying, we find that the horizontal displacement is equal to:
Horizontal displacement ≈ 76,951.083 m

Use conservation of momentum to find the initial velocity of the block/bullet.

Now using that velocity, find its intial KE. If you set that KE = mgh, h is the max vertical height.

Now draw the figure, find out at h what the angle the pendulum is, and from that, the horizontal movement.