A 0.010 kg bullet is fired into a 7 kg ballistic pendulum. The pendulum rises 0.08m before it stops. Use conservation of energy and momentum to find the initial velocity of the bullet.
To find the initial velocity of the bullet, we can use both the conservation of energy and conservation of momentum principles.
First, let's consider the conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision. In this case, the bullet is initially moving and the pendulum is at rest, so we can write:
(mass of the bullet) * (initial velocity of the bullet) = (mass of the bullet + mass of the pendulum) * (final velocity of the bullet and the pendulum)
Using the given values, we can substitute the masses into this equation:
(0.010 kg) * (initial velocity of the bullet) = (0.010 kg + 7 kg) * (final velocity of the bullet and the pendulum)
Since the pendulum rises after the collision, the final velocity of the bullet and the pendulum can be considered as zero. Therefore, the equation simplifies to:
(0.010 kg) * (initial velocity of the bullet) = (0.010 kg + 7 kg) * 0
The momentum conservation equation tells us that the initial velocity of the bullet is zero, which means the bullet isn't moving when it hits the pendulum. However, this conclusion seems incorrect, so let's consider the conservation of energy to find where the mistake might be.
According to the principle of conservation of energy, the initial kinetic energy of the bullet is equal to the sum of the final kinetic energy of the bullet and the potential energy of the pendulum after the collision. Mathematically, this can be expressed as:
(1/2) * (mass of the bullet) * (initial velocity of the bullet)^2 = (1/2) * (mass of the bullet) * (final velocity of the bullet)^2 + (mass of the pendulum) * (acceleration due to gravity) * (height gained by the pendulum)
Substituting the given values, we have:
(1/2) * (0.010 kg) * (initial velocity of the bullet)^2 = 0 + (7 kg) * (9.8 m/s^2) * (0.08 m)
Simplifying the equation further:
(0.005 kg) * (initial velocity of the bullet)^2 = 5.6 J
Now, to solve for the initial velocity of the bullet, we can divide both sides of the equation by (0.005 kg) and then take the square root of both sides:
(initial velocity of the bullet)^2 = (5.6 J) / (0.005 kg)
(initial velocity of the bullet)^2 = 1120 m^2/s^2
Taking the square root:
initial velocity of the bullet = √(1120 m^2/s^2) ≈ 33.54 m/s
So, the initial velocity of the bullet is approximately 33.54 m/s.