How Fast Was the Bullet That Hit the Pendulum?

[candycooke]

A 12.5 g bullet is shot into a ballistic pendulum that has a mass of 2.37 kg. The pendulum rises a distance of 9.55 cm above its resting position. What was the speed of that bullet?
m(bullet) = 0.0125 kg
m(pendulum) = 2.37 kg
h = 0.0955 m

I know that I'm supposed to use conservation of energy:
P(i)=P(f)
m(1)v(i)=(m(1)+m(2))v(f)

Unfortunately this equation does not take into account how high the pendulum swung, which is obviously an important factor.

 

[Doc Al]

Treat this problem as having two parts:
(1) The collision of bullet and pendulum. (What's conserved here?)
(2) The swinging of pendulum+bullet after the collision. (What's conserved here?)

 

[Moetasim]

To calculate the speed of the bullet, we need to use the conservation of momentum and energy equations together.

First, we can use the conservation of momentum equation, which states that the initial momentum of the bullet (m(bullet)v(i)) is equal to the final momentum of the bullet and pendulum system (m(bullet)v(f) + m(pendulum)v(f)). This can be written as:

m(bullet)v(i) = (m(bullet) + m(pendulum))v(f)

We can also use the conservation of energy equation, which states that the initial kinetic energy of the bullet (1/2m(bullet)v(i)^2) is equal to the final kinetic energy of the bullet and pendulum system (1/2(m(bullet) + m(pendulum))v(f)^2) plus the potential energy gained by the pendulum (m(pendulum)gh). This can be written as:

1/2m(bullet)v(i)^2 = 1/2(m(bullet) + m(pendulum))v(f)^2 + m(pendulum)gh

Now, we can use algebra to solve for v(f) in both equations and set them equal to each other, since they are equal in both equations. This gives us:

m(bullet)v(i) = (m(bullet) + m(pendulum))v(f)
1/2m(bullet)v(i)^2 = 1/2(m(bullet) + m(pendulum))v(f)^2 + m(pendulum)gh

Solving for v(f) in the first equation gives us:

v(f) = m(bullet)v(i) / (m(bullet) + m(pendulum))

Substituting this into the second equation and solving for v(i) gives us:

v(i) = √(2gh(m(bullet) + m(pendulum)) / m(bullet))

Plugging in the given values, we get a speed of approximately 203 m/s for the bullet. So, the speed of the bullet in this scenario is 203 m/s.