Inelastic Collision and Conservation of Momentum

[mexicandeligh]

Homework Statement

A student performs a ballistic pendulum experiment using an apparatus similar to that shown it the figure. Initially the bullet is fired at the block while the block is at rest (at its lowest swingin point). After the bullet hits the block, the block rises to its highest position, see dashed block in the figure, and continues swinging back and forth.
The following data is obtained:
the maximum height of the pendulum rises 3 cm,
at the maximum height the pendulum subtends an angle of 36.9,
the mass of the bullet is 64 g, and
the mass of the pendulum bob is 889 g.
The acceleration of gravity is 9.8 m/s^2
Determine the initial speed of the projectile
Answer in units of m/s

 

[Nebozilla]

Have you tried it? Try using conservation of energy and momentum.

 

[kerimek]

The initial speed of the projectile can be determined using the principles of inelastic collision and conservation of momentum. In an inelastic collision, the objects involved stick together and move as one unit after the collision. Therefore, the momentum of the bullet before the collision is equal to the momentum of the combined bullet and block after the collision.

We can use the conservation of momentum equation, P1 = P2, where P1 is the initial momentum and P2 is the final momentum. The initial momentum of the bullet is calculated as m1v1, where m1 is the mass of the bullet and v1 is its initial velocity. The final momentum of the combined bullet and block is calculated as (m1 + m2)v2, where m2 is the mass of the block and v2 is the final velocity of the combined system. Since the block was initially at rest, its initial velocity is zero, and we can rewrite the equation as m1v1 = (m1 + m2)v2.

Substituting the given values, we get (0.064 kg)v1 = (0.064 kg + 0.889 kg)v2. Simplifying, we get v1 = 13.91v2.

Next, we can use the conservation of energy to relate the initial kinetic energy of the bullet to the potential energy of the pendulum at its maximum height. The initial kinetic energy of the bullet is calculated as 1/2m1v1^2 and the potential energy of the pendulum at its maximum height is calculated as mgh, where m is the mass of the pendulum bob, g is the acceleration due to gravity, and h is the maximum height reached by the pendulum.

Substituting the given values, we get 1/2(0.064 kg)v1^2 = (0.889 kg)(9.8 m/s^2)(0.03 m). Simplifying, we get v1^2 = 4.244 m^2/s^2.

Finally, we can substitute this value of v1^2 into our earlier equation v1 = 13.91v2 to get (4.244 m^2/s^2) = 13.91v2. Solving for v2, we get v2 = 0.305 m/s.

Therefore, the initial speed of the projectile is 0.305 m/s.