In a ballistic pendulum experiment, a small marble is fired into a cup attached to the end of a pendulum. If the mass of the marble is 0.0255 kg and the mass of the pendulum is 0.250 kg, how high will the pendulum swing if the marble has an initial speed of 5.75 m/s? Assume that the mass of the pendulum is concentrated at its end so that linear momentum is conserved during this collision.

To solve this problem, we can use the principle of conservation of linear momentum. According to this principle, the total momentum before collision is equal to the total momentum after collision.

Step 1: Calculate the initial momentum of the marble.
The initial momentum (p₁) of an object can be calculated using the formula:
p₁ = m₁ * v₁
Where:
m₁ = mass of the object (0.0255 kg)
v₁ = initial speed of the marble (5.75 m/s)

p₁ = 0.0255 kg * 5.75 m/s
p₁ = 0.1466 kg·m/s

Step 2: Calculate the final momentum.
Since linear momentum is conserved during the collision, the final momentum (p₂) will be the same as the initial momentum.
p₂ = p₁
p₂ = 0.1466 kg·m/s

Step 3: Calculate the final speed of the system.
The final speed (v₂) can be calculated using the formula:
p₂ = m₂ * v₂
Where:
m₂ = combined mass of the marble and pendulum (0.0255 kg + 0.250 kg = 0.275 kg)

v₂ = p₂ / m₂
v₂ = 0.1466 kg·m/s / 0.275 kg
v₂ ≈ 0.5327 m/s

Step 4: Calculate the potential energy of the system at the maximum height.
At the highest point of the swing, the kinetic energy is zero and all the energy is potential energy.

The potential energy (PE) can be calculated using the formula:
PE = m * g * h
Where:
m = combined mass of the system (0.0255 kg + 0.250 kg = 0.275 kg)
g = acceleration due to gravity (9.8 m/s²)
h = height

PE = m * g * h
0.275 kg * 9.8 m/s² * h = 0.275 kg * g * h

At the highest point of the swing, all the kinetic energy of the system is converted into potential energy:
(1/2) * (m * v₂²) = PE

(1/2) * (0.275 kg * (0.5327 m/s)²) = 0.275 kg * 9.8 m/s² * h

Step 5: Solve for the height (h).
0.5 * 0.275 kg * (0.5327 m/s)² = 0.275 kg * 9.8 m/s² * h

0.0735 kg·m²/s² = 2.695 kg·m²/s² * h

h = 0.0735 kg·m²/s² / (2.695 kg·m²/s²)
h ≈ 0.0273 meters

Step 6: Convert the height to centimeters (optional).
1 meter = 100 centimeters
0.0273 meters ≈ 2.73 centimeters

Therefore, the pendulum will swing to a height of approximately 2.73 centimeters.

To solve this problem, we can use the principle of conservation of linear momentum. The initial linear momentum of the system (marble + pendulum) before the collision is equal to the final linear momentum after the collision.

The linear momentum (p) is equal to the product of mass (m) and velocity (v).

Before the collision:
Momentum of the marble = mass of the marble * initial velocity of the marble
= 0.0255 kg * 5.75 m/s

After the collision:
Momentum of the marble and pendulum = (mass of the marble + mass of the pendulum) * final velocity of the system

Since linear momentum is conserved, we can equate these two expressions and solve for the final velocity of the system.

(mass of the marble * initial velocity of the marble) = (mass of the marble + mass of the pendulum) * final velocity of the system

Now, we can find the final velocity of the system by rearranging the equation:

final velocity of the system = (mass of the marble * initial velocity of the marble) / (mass of the marble + mass of the pendulum)

Substituting the given values:
final velocity of the system = (0.0255 kg * 5.75 m/s) / (0.0255 kg + 0.250 kg)

Now that we have the final velocity of the system, we can find the height to which the pendulum will swing using the conservation of mechanical energy.

The mechanical energy before the collision (consisting of only the kinetic energy of the marble) will be equal to the mechanical energy after the collision (consisting of the potential energy of the pendulum at its maximum height).

The mechanical energy before the collision is given by:
Kinetic energy of the marble = (1/2) * mass of the marble * (initial velocity of the marble)^2

The mechanical energy after the collision is given by:
Potential energy of the pendulum = mass of the pendulum * gravitational acceleration * height

Since the mechanical energy is conserved, we can equate these two expressions and solve for the height:

(1/2) * mass of the marble * (initial velocity of the marble)^2 = mass of the pendulum * gravitational acceleration * height

Now, we can find the height by rearranging the equation:

height = ((1/2) * (mass of the marble * (initial velocity of the marble)^2)) / (mass of the pendulum * gravitational acceleration)

Substituting the given values:
height = ((1/2) * (0.0255 kg * (5.75 m/s)^2)) / (0.250 kg * 9.8 m/s^2)

Calculating this expression will give the answer to the question.

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