a-28-g rifle bullet traveling 230m/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.
a similar question from the past. https://www.jiskha.com/questions/426977/A-25-rifle-bullet-traveling-280-buries-itself-in-a-3-8-pendulum-hanging-on-a-3-0
To determine the vertical and horizontal components of the pendulum's displacement, we can use the conservation of momentum and energy.
1. Conservation of Momentum:
The initial momentum of the bullet is equal to the final momentum of the bullet and pendulum system after the collision.
Initial momentum of the bullet = mass of bullet * velocity of bullet
Final momentum of bullet and pendulum = (mass of bullet + mass of pendulum) * final velocity
Since the bullet buries itself in the pendulum, the final velocity is zero.
Therefore,
mass of bullet * velocity of bullet = (mass of bullet + mass of pendulum) * 0
mass of bullet * velocity of bullet = mass of bullet * 0 + mass of pendulum * 0
mass of bullet * velocity of bullet = 0
This equation tells us that the momentum of the bullet is zero after the collision.
2. Conservation of Energy:
The bullet's kinetic energy before the collision is converted into potential energy of the pendulum after the collision.
Initial kinetic energy of the bullet = (1/2) * mass of bullet * velocity of bullet^2
Final potential energy of the pendulum = mass of pendulum * g * height
Since the bullet buries itself in the pendulum, all the kinetic energy is converted into potential energy. The height of the pendulum can be found using trigonometry:
Height = Length of the pendulum * (1 - cos(angle)), where the angle is the maximum angle reached by the pendulum.
3. Horizontal and Vertical Displacement:
Using the height obtained from step 2, the vertical and horizontal components of the pendulum's displacement can be calculated.
Vertical displacement = Height = Length of the pendulum * (1 - cos(angle))
Horizontal displacement = Length of the pendulum * sin(angle)
Note: To find the maximum angle reached by the pendulum, more information is needed, such as the angle at which the bullet hits the pendulum.
To determine the vertical and horizontal components of the pendulum's displacement, we first need to analyze the situation and apply the laws of physics.
Given:
- Rifle bullet mass (m1) = 28 g = 0.028 kg
- Rifle bullet velocity (v1) = 230 m/s
- Pendulum mass (m2) = 3.6 kg
- Pendulum string length (L) = 2.8 m
Horizontal Component of Displacement:
The horizontal component remains unchanged as the bullet is buried in the pendulum. Therefore, the horizontal component of the pendulum's displacement remains zero.
Vertical Component of Displacement:
Initially, the pendulum is at rest. When the bullet buries itself in the pendulum, it imparts momentum to it. According to the conservation of momentum, the total momentum before and after the collision must be the same.
The momentum of the bullet before the collision is given by:
Momentum (p1) = mass (m1) × velocity (v1)
The momentum of the pendulum after the collision is given by:
Momentum (p2) = mass (m2) × velocity (v2)
Since the pendulum was initially at rest, the pendulum's velocity (v2) can be assumed to be zero.
Therefore, we can calculate the velocity (v2) of the pendulum after the collision using the law of conservation of momentum:
p1 = p2
m1 × v1 = m2 × v2
0.028 kg × 230 m/s = 3.6 kg × v2
v2 = (0.028 kg × 230 m/s) / 3.6 kg
v2 ≈ 1.8 m/s
Now, we can analyze the motion of the pendulum after the collision. As the pendulum swings upward, it follows an arc. At the highest point of the arc, the velocity of the pendulum momentarily becomes zero. Therefore, the vertical component of the pendulum's displacement is equal to the maximum height reached.
Using the conservation of mechanical energy, we can find the maximum height (h) reached by the pendulum:
Potential Energy (PE) at the maximum height = Kinetic Energy (KE) before the collision
PE = m2 × g × h (where g is the acceleration due to gravity)
KE = (1/2) × m2 × v2^2
Setting these two equations equal:
m2 × g × h = (1/2) × m2 × v2^2
h = (1/2) × v2^2 / g
Substituting the values:
h = (1/2) × (1.8 m/s)^2 / 9.8 m/s^2
h ≈ 0.164 m
Therefore, the vertical component of the pendulum's displacement is approximately 0.164 meters upwards, and the horizontal component of the displacement is zero.