A 14 g bullet is fired into the bob of a ballistic pendulum of mass 1.6 kg. When the bob is at its maximum height, the strings make an angle of 60° with the vertical. The length of the pendulum is 2.3 m. Find the speed of the bullet.
?m/s
To find the speed of the bullet, we can use the principle of conservation of momentum and conservation of energy.
First, let's calculate the initial speed of the bullet before it hits the pendulum. We know the mass of the bullet is 14 g (or 0.014 kg).
Let's assume the velocity of the bullet before it hits the pendulum is v.
Using the principle of conservation of momentum, the momentum before the collision is equal to the momentum after the collision.
The momentum before the collision is given by the formula:
m1 * v1 = (mass of the bullet) * (initial velocity of the bullet)
The momentum after the collision is given by the formula:
(mass of the bullet + mass of the pendulum) * (final velocity of the pendulum)
Since the bullet is lodged into the pendulum, the final velocity of the pendulum is zero. Therefore, we have:
(mass of the bullet + mass of the pendulum) * 0 = (mass of the bullet) * (initial velocity of the bullet)
Now, we can solve for the initial velocity of the bullet:
v = 0 / (mass of the bullet + mass of the pendulum)
v = 0
Therefore, the initial velocity of the bullet is 0 m/s.
Next, let's calculate the final velocity of the pendulum bob right at its maximum height.
At the maximum height, the bob is momentarily at rest, and all of the initial kinetic energy of the bullet is converted into potential energy of the pendulum bob.
The potential energy of the pendulum bob can be calculated using the formula:
Potential energy = mass * gravitational acceleration * height
Since the height of the bob at the maximum height is the length of the pendulum (2.3 m) and the angle between the string and the vertical is 60 degrees, we can find the height by taking the vertical component of the length:
height = length * cos(angle)
height = 2.3 m * cos(60 degrees)
Now, we can calculate the potential energy:
Potential energy = (mass of the pendulum) * 9.8 m/s^2 * height
Potential energy = 1.6 kg * 9.8 m/s^2 * (2.3 m * cos(60 degrees))
Finally, we can use the principle of conservation of energy to equate the initial kinetic energy of the bullet to the final potential energy of the pendulum:
Initial kinetic energy of the bullet = Potential energy of the pendulum bob
(1/2) * (mass of the bullet) * (initial velocity of the bullet)^2 = (mass of the pendulum) * 9.8 m/s^2 * height
Substituting the known values:
(1/2) * (0.014 kg) * (0)^2 = (1.6 kg) * 9.8 m/s^2 * (2.3 m * cos(60 degrees))
After simplification, we find:
0 = 35.256 kg * m^2/s^2 * cos(60 degrees)
Since cos(60 degrees) = 0.5, the equation simplifies to:
0 = 35.256 kg * m^2/s^2 * 0.5
0 = 17.628 kg * m^2/s^2
Since the left side of the equation is zero, the right side must also be zero.
Therefore, the speed of the bullet is 0 m/s.
Hence, the speed of the bullet is 0 m/s.