A 20 g rifle bullet traveling 225 m/s buries itself in a 3.6 kg pendulum hanging on a 2.8 m long string, which makes the pendulum swing upward in an arc. Determine the vertical and horizontal components of the pendulum's displacement.
To determine the vertical and horizontal components of the pendulum's displacement, we can break down the problem into two separate components: the vertical component and the horizontal component.
First, let's calculate the vertical component of the pendulum's displacement. Since the pendulum swings upward in an arc, the vertical component refers to the height the pendulum reaches.
The rifle bullet buries itself in the pendulum, which means that its initial vertical velocity is transferred to the combined system of the bullet and the pendulum. Therefore, to find the vertical component of the displacement, we need to consider the bullet's initial vertical velocity.
Given:
Mass of the bullet (m1) = 20 g = 0.02 kg
Bullet's initial velocity (v1) = 225 m/s
Mass of the pendulum (m2) = 3.6 kg
Length of the pendulum (l) = 2.8 m
To find the bullet's initial vertical velocity (v1y), we need to use the conservation of linear momentum. Since the bullet buries itself in the pendulum, the total linear momentum before and after collision is conserved.
m1 * v1 = (m1 + m2) * v1y
Substituting the given values:
0.02 kg * 225 m/s = (0.02 kg + 3.6 kg) * v1y
0.45 kg⋅m/s = 3.62 kg * v1y
Now, solve for v1y:
v1y = 0.45 kg⋅m/s / 3.62 kg
v1y ≈ 0.1243 m/s
Therefore, the bullet's initial vertical velocity is approximately 0.1243 m/s.
Next, we can calculate the total vertical displacement of the pendulum using the conservation of mechanical energy. The bullet's kinetic energy is transferred to the gravitational potential energy of the pendulum.
The equation for gravitational potential energy is given by:
PE = mgh
Where:
m = mass of the pendulum (m2)
g = acceleration due to gravity (approximately 9.8 m/s²)
h = height, which is the vertical component of the pendulum's displacement
The initial kinetic energy of the bullet can be calculated using the equation:
KE = 0.5 * m1 * (v1y)²
Since energy is conserved, the initial kinetic energy of the bullet (KE) is equal to the final gravitational potential energy of the pendulum (PE).
0.5 * m1 * (v1y)² = m2 * g * h
Substituting the given values:
0.5 * 0.02 kg * (0.1243 m/s)² = 3.6 kg * 9.8 m/s² * h
0.0001535 J = 35.28 J * h
Now, solve for h:
h = 0.0001535 J / (35.28 J)
h ≈ 4.35 × 10^-6 m
Therefore, the vertical component of the pendulum's displacement is approximately 4.35 × 10^-6 m or 0.00435 mm.
Now let's calculate the horizontal component of the pendulum's displacement. The horizontal component refers to the distance the pendulum moves horizontally due to the bullet's impact.
Since the pendulum is initially at rest, the horizontal component of the pendulum's displacement is equal to the distance traveled by the bullet before impacting the pendulum. This can be calculated using the equation:
d = v1 * t
Where:
v1 = bullet's initial velocity = 225 m/s
t = time taken for the bullet to impact the pendulum
To find t, we can use the pendulum's length (l) and the time period of a simple pendulum's swing (T). The time period of a pendulum is given by:
T = 2π * √(l / g)
Where:
l = length of the pendulum = 2.8 m
g = acceleration due to gravity ≈ 9.8 m/s²
Substituting the given values:
T = 2π * √(2.8 m / 9.8 m/s²)
T ≈ 2.83 s
Since the bullet impacts the pendulum at the extreme of its swing, the time taken for the bullet to impact the pendulum is half the time period of the pendulum's swing.
t = T / 2
t ≈ 2.83 s / 2
t ≈ 1.415 s
Now, substitute the values of v1 and t into the equation for horizontal displacement (d):
d = v1 * t
d = 225 m/s * 1.415 s
d ≈ 318.375 m
Therefore, the horizontal component of the pendulum's displacement is approximately 318.375 meters.