A ballistic pendulum was a device used in the past century to measure the speed of bullets. The pendulum consists of a large block of wood suspended from long wires. Initially, the pendulum is at rest. The bullet strikes the block horizontally and remains stuck in it. The impact of the bullet puts the block in motion, causing it to swing upward to a height h. If the bullet has a mass of 8.90 g, and the block of mass 2.20 kg swings up to a height of h = 7.70x101 cm, what was the speed of the bullet (in m/s) before impact ?
To find the speed of the bullet before impact, we can start by considering the conservation of momentum and conservation of energy principles.
1. Conservation of Momentum:
Before impact, the bullet and the pendulum are separate objects, so we can write:
Initial momentum of bullet = Final momentum of bullet + Final momentum of pendulum
The initial momentum of the bullet is given by:
Initial momentum of bullet = mass of the bullet x initial velocity of the bullet
Since the bullet remains stuck in the pendulum, the final momentum of the bullet is zero:
Final momentum of bullet = mass of the bullet x final velocity of the bullet = 0
The final momentum of the pendulum can be calculated using its mass and final velocity. However, we don't know the final velocity yet, so we need to calculate it using the conservation of energy.
2. Conservation of Energy:
When the pendulum swings upward, it gains gravitational potential energy at the highest point. This energy comes from the initial kinetic energy of the system (bullet + pendulum). We can express this as:
Initial kinetic energy of the system = Final potential energy of the pendulum
The initial kinetic energy of the system is given by:
Initial kinetic energy of the system = (1/2) x mass of the bullet x (initial velocity of the bullet)^2
The final potential energy of the pendulum is given by:
Final potential energy of the pendulum = mass of the pendulum x acceleration due to gravity x height (h) reached
Now, we can equate the initial kinetic energy of the system with the final potential energy of the pendulum and solve for the initial velocity of the bullet.
(1/2) x mass of the bullet x (initial velocity of the bullet)^2 = mass of the pendulum x acceleration due to gravity x height (h)
Now plug in the given values:
(1/2) x 8.90 g x (initial velocity of the bullet)^2 = 2.20 kg x 9.81 m/s^2 x 7.70x10^-2 m
Simplifying the equation and solving for the initial velocity of the bullet:
Initial velocity of the bullet = sqrt((2 x mass of the pendulum x acceleration due to gravity x height) / mass of the bullet)
Plugging in the values:
Initial velocity of the bullet = sqrt((2 x 2.20 kg x 9.81 m/s^2 x 7.70x10^-2 m) / 8.90 g)
Finally, calculate the result to find the speed of the bullet before impact in m/s.