A 5 gram bullet is fired into a 7 kg ballistic pendulum suspended vertically from the ceiling. If the bullet embeds into the block and the two move as one to a new final height of 12 cm. What was the initial velocity of the bullet before impacting the pendulum?
The moment when the bullet collides with the block, is the max of kinetic energy of the system (bullet embeds block). When the system swings to the maximum height = 12cm, it's when kinetic energy converts to potential energy. So that, Max-of-kinetic-energy = Max-of-potential-energy
=> 0.5*(m+M)*V^2 = (m+M)*g*h
=> V = sqr(2gh)
Apply the rule of conservation of momentum, we have:
m*v + M*0 = (m+M)*V
=> v = [(m+M)*V]/m
To determine the initial velocity of the bullet, we can make use of the principle of conservation of momentum and the principle of conservation of mechanical energy.
1. Conservation of Momentum:
The momentum of a system remains constant before and after a collision if no external forces act on it. Mathematically, this can be expressed as:
m1v1 + m2v2 = (m1 + m2)V
where
- m1 is the mass of the bullet,
- v1 is the initial velocity of the bullet,
- m2 is the mass of the pendulum,
- v2 is the initial velocity of the pendulum, and
- V is the final velocity of the combined bullet and pendulum.
Since the pendulum is initially at rest (v2 = 0) in the vertical position, the equation simplifies to:
m1v1 = (m1 + m2)V
2. Conservation of Mechanical Energy:
The mechanical energy (kinetic energy + potential energy) of the system remains constant before and after the collision if no external forces (such as friction or air resistance) are acting on it. Mathematically, this can be expressed as:
(1/2)m1v1^2 + m2gh2 = (1/2)(m1 + m2)V^2
where
- h2 is the final height of the combined bullet and pendulum, and
- g is the acceleration due to gravity.
We can now solve these two equations simultaneously to find the initial velocity of the bullet.
Given:
- Mass of the bullet (m1) = 5 grams = 0.005 kg
- Mass of the pendulum (m2) = 7 kg
- Final height (h2) = 12 cm = 0.12 m
- Acceleration due to gravity (g) = 9.8 m/s^2
Substituting these values into the equations, we get:
(0.005 kg)v1 = (0.005 kg + 7 kg)V --> Equation 1
(1/2)(0.005 kg)v1^2 + 7 kg * 9.8 m/s^2 * 0.12 m = (1/2) * (0.005 kg + 7 kg)V^2 --> Equation 2
Now, we can solve Equation 1 for V:
V = (0.005 kg)v1 / (0.005 kg + 7 kg)
Substitute this value of V into Equation 2:
(1/2)(0.005 kg)v1^2 + 7 kg * 9.8 m/s^2 * 0.12 m = (1/2)(0.005 kg + 7 kg)((0.005 kg)v1 / (0.005 kg + 7 kg))^2
Simplifying and solving for v1 will give us the initial velocity of the bullet before impacting the pendulum.