### Purpose

To utilize two different methods of determining the initial velocity of a fired ball, namely a ballistic pendulum and treating the ball as a projectile, and then compare these two calculated values. The loss of kinetic energy from firing the ball into the pendulum is also an area of interest.

### Hypothesis

The initial velocity determined by firing the ball into the ballistic pendulum should theoretically be equal to the initial velocity determined by firing the ball as a projectile.

### Labeled Diagrams

See attached sheet.

### Data

**Part One**

Trial | p | y |

1 | 31 | 0.160 |

2 | 30 | 0.159 |

3 | 29 | 0.158 |

4 | 29 | 0.158 |

5 | 30 | 0.159 |

m = 0.0697 kg | M= 0.2785 kg | y _{1}= 0.0788 m |

Trial | y | V (m/s) | v (m/s) |

1 | 0.0812 | 1.26 | 6.29 |

2 | 0.0802 | 1.25 | 6.24 |

3 | 0.0792 | 1.24 | 6.19 |

4 | 0.0792 | 1.24 | 6.19 |

5 | 0.0802 | 1.25 | 6.24 |

v= 6.23 m/s α_{v}=0.0187 m/s

**Part Two**

Trial | X (m) | v(m/s) |

1 | 2.69 | 8.41 |

2 | 2.75 | 8.56 |

3 | 2.91 | 9.10 |

4 | 2.82 | 8.81 |

5 | 2.83 | 8.85 |

Y=1.003 m v= 8.75 m/s α_{v}=0.120 m/s

### Questions

**1. Compare the two different values of v average. Calculate the percent difference between them. State whether the two measurements agree within the combined standard errors of the two values of v average.**

The average initial velocity for the ballistic pendulum was 6.23 m/s while the average initial velocity for the projectile determination was 8.75 m/s. This is a percent difference of 33.6%. It should have been expected that these two values would be equal. The two measurements also do not agree within the combined standard errors of the two values for v average, as the standard errors only total 0.1387 m/s, and the average velocities fall out of that range.

**2. Calculate the loss in kinetic energy when the ball collides with the pendulum as the difference between ½ mv ^{2} (the kinetic energy before) and ½ (m + M)V^{2} (the kinetic energy immediately after the collision). What is the fractional loss in kinetic energy? Calculate by dividing the loss by the original kinetic energy.**

The average kinetic energy before the collision is 1.35 J and the average kinetic energy immediately after the collision is 0.272 J, so the loss of kinetic energy is 1.08 J. The fractional loss in kinetic energy is 0.8.

**3. Calculate the ratio M / (m + M) for the values of m and M in Part 1. Compare this ratio with the ratio calculated in the previous question. Express the fractional loss of kinetic energy in symbol form and use equations from the Theory section to show it should equal M / (m + M). **

The ratio M / (m + M) is equal to 0.8. This ratio is exactly the same as the fractional loss of kinetic energy.

The fractional loss of kinetic energy equals ( ½ mv^{2} – ½ (m + M)V^{2 }) / ( ½ mv^{2 }).

### Conclusion

During part one of the experiment, a ball was fired into a ballistic pendulum to ultimately determine its initial velocity. This process was repeated five times in order to obtain average values to work with in order to eliminate error. By massing the ball and the pendulum, recording the initial and final heights, the values for V and finally v could be calculated. It was found that the average initial velocity of the ball was 6.23 m/s.

During part two of the experiment, the same ball was fired as a projectile instead of into a ballistic pendulum. The ball was fired from a table horizontally to the ground. A piece of carbon paper was used to capture the spot where the ball first struck the ground. Height and horizontal distance the ball traveled were then measured in order to determine the initial velocity of the ball. The average initial velocity of the ball was 8.75 m/s.

As far as the accuracy of the results from the lab, the percent difference between the average velocities calculated is 33.6%. This is a fairly significant difference, which suggests that there sources of error during the procedure. The notched part of the ballistic setup could have had finer groves to yield more accurate measurements. The major contributor of error, however, was most likely from the distance measurements from the projectile part of the lab. One positive to come from the results was that the fractional loss in kinetic energy was identical to the mass ratios from the ballistic pendulum setup, which is theoretically expected.

### Equations

∆KE = ½ mv^{2}

½ (m + M)V^{2} = (m + M)gh

mv = (m + M)V

V = (2gh)^{0.5} v = (m + M) (2gh)^{0.5} / m v = ∆x / (2∆y / g)^{0.5}