Ballistic pendulum

Objective:

• Determine the velocity of a bullet by firing it into a suspended block and also figure out the propagated uncertainty in that velocity.

Base Theory:

1. The ballistic pendulum is a example of a dissipative collision in which conservation of momentum can be used for analysis.
2. The moment when the ball is fired into a suspended block, the energy is not conserved because the energy may go into inaccessible forms such as internal energy or chemical energy we don't actually know or doesn't have an equation to calculate them yet.
3. After the collision, conservation of energy can be used in the swing of the combined masses upward, since the gravitational potential energy is conservative.
4. We will use both equation from conservation of momentum during collision, and conservation of energy after collision to determine the initial velocity of the bullet.

Set-up:

• The apparatus uses a nylon pendulum with a tapered rubber insert. Both pendulum and base have leveling screws.
• The spring-loaded gun has a self-locking trigger. The scale is marked in degrees. 

Below is the apparatus of the experiment:

Figure : The apparatus of the experiment.

Performing the experiment:

1. First, loading the ball into the gun. Then set the self-locking trigger of the gun ready.
2. Let the trigger go, the ball is fired into the block and makes the block raise to a certain height.
3. Measuring appropriate data and perform calculation to figure out the initial velocity.

Data Collection and Analysis:

• Collecting necessary data for calculation:
• Mass of block = 80.9g ± 0.1g
• Mass of ball = 7.63g ± 0.1g
• Length of string = 21.7 cm ± 0.1cm
• Measuring the angle after collision

Below is the diagram of the experiment:

Figure : The diagram shows how the experiment looks like

• We apply the conservation of momentum before and during collision and conservation of energy after collision to figure out the initial velocity of the ball. Figure shows how to derive an equation to calculate the initial velocity 

Figure : Deriving an equation to calculate initial velocity

• Since we already measured masses of ball and block, the only thing we need to figure out is the height of the object. In other words, we need to calculate the height and its propagated uncertainty.
• Deriving an equation and calculate the height.

Figure : Calculating the height of the object when it swings
h = 9.7 * 10^-3 m
• Calculating the propagated uncertainty of the height.

Figure : Calculating the propagated uncertainty of the height

dh = 4.58 * 10^-3 m

• Since we have all data we want, we just need to plug in back to the velocity's equation to figure out the initial velocity

Figure : Plugging in data and calculating the initial velocity of the ball

v = 5.1 m/s

• We also calculate the propagated uncertainty of the initial velocity. Work is shown in figure 

Figure : Calculating the propagated uncertainty of initial velocity

dv = 0.502m/s

Conclusion:

• In this experiment, we base on the theory of conservation of momentum and energy to figure out the initial velocity of the ball. 
• The conservation of momentum is applied before and during the collision because there is no external force.
• The conservation of energy is applied after collision to figure out the relationship between the initial velocity of the ball and the height of the whole system.
• Conservation of energy is not applied during collision because the energy may transfer to some inaccessible forms such as chemical energy or internal energy. 
• The final result we got for initial velocity of the ball is v = 5.1 m/s ± 0.502m/s