A 0.02-kg bullet collides with a 5.75-kg pendulum. After the
collision, the bullet remains embedded onto the pendulum and the pair swings up to a maximum height of 0.386m.
1. Find the total energy of the pair at the maximum height of 0.386 m.
2.What is the initial speed of the pair at the bottom before swinging? Note: energy is conserved from the bottom to the maximum height.
3.What is the initial speed of the bullet? Note: In the collision of bullet and pendulum, momentum is conserved.
1.p
To solve these questions, we will use principles of conservation of energy and momentum.
1. Finding the total energy of the pair at the maximum height:
To find the total energy, we need to consider the kinetic energy and potential energy of the system. At the maximum height, all the initial kinetic energy of the pair is converted into potential energy.
The formula for potential energy is: Potential Energy (PE) = mass * acceleration due to gravity * height
In this case, the total mass of the system is the mass of the bullet and the mass of the pendulum: 0.02 kg + 5.75 kg = 5.77 kg.
Using the given values, we can calculate the potential energy:
PE = 5.77 kg * 9.8 m/s² * 0.386 m
= 21.8352 Joules
Therefore, the total energy of the pair at the maximum height is 21.8352 Joules.
2. Finding the initial speed of the pair at the bottom before swinging:
Since energy is conserved from the bottom to the maximum height, we can equate the initial kinetic energy at the bottom with the potential energy at the maximum height.
The formula for kinetic energy is: Kinetic Energy (KE) = 0.5 * mass * speed²
Using the given values, we know that the potential energy at the maximum height (21.8352 J) is equal to the initial kinetic energy at the bottom.
21.8352 J = 0.5 * 5.77 kg * speed²
Simplifying the equation:
43.6704 J = 5.77 kg * speed²
Dividing both sides by 5.77 kg:
7.556 m²/s² = speed²
Taking the square root of both sides, we can solve for the speed:
speed = √(7.556 m²/s²)
≈ 2.75 m/s
Therefore, the initial speed of the pair at the bottom before swinging is approximately 2.75 m/s.
3. Finding the initial speed of the bullet:
In the collision of the bullet and the pendulum, momentum is conserved. Since the bullet remains embedded onto the pendulum, we can equate the momentum before and after the collision.
The formula for momentum is: Momentum (p) = mass * velocity
Using the given values, we can set up the equation:
(0.02 kg)(initial speed of the bullet) = (5.75 kg)(final velocity of the bullet and pendulum)
Since the bullet remains embedded onto the pendulum, their final velocity will be the same. Therefore, we can rewrite the equation as:
(0.02 kg)(initial speed of the bullet) = (5.75 kg)(final velocity of the bullet and pendulum)
We can rearrange the equation to solve for the initial speed of the bullet:
(initial speed of the bullet) = (5.75 kg)(final velocity of the bullet and pendulum) / (0.02 kg)
To find the final velocity of the bullet and pendulum, let's consider the conservation of momentum from the initial state (before the collision) to the final state (after the collision):
(0.02 kg)(initial speed of the bullet) + (5.75 kg)(initial velocity of the pendulum) = (5.77 kg)(final velocity of the bullet and pendulum)
Since the bullet and the pendulum are initially at rest, the initial velocity of the pendulum is 0. Therefore, we can simplify the equation to:
(0.02 kg)(initial speed of the bullet) = (5.77 kg)(final velocity of the bullet and pendulum)
Substituting this equation into the initial speed equation:
(initial speed of the bullet) = (5.75 kg)(final velocity of the bullet and pendulum) / (0.02 kg)
= (5.75 kg)(0) / (0.02 kg)
= 0 m/s
Therefore, the initial speed of the bullet is 0 m/s, as it starts from rest before the collision with the pendulum.