Study of the Ballistic Pendulum

Introduction:

The study of the ballistic pendulum began with Benjamin Robins in the early 1700's. Robins was an English mathematician and writer on ballistics, and used this testing apparatus for the evaluation of the strength of gunpowder and the measurement of air resistance (Calvert, 2006). At the time of the American Revolution a loyalist named Benjamin Thompson also utilized and expanded the use to the ballistic pendulum. He further evaluated the efficiency of gunpowder. He fired from a small cannon, into a heavy suspended target and found the speed of the bullet by finding the rise of the target (Stern, 2006). The study of the ballistic pendulum is quite unique and incorporates many important physical laws. For this reason, peers interested in physics and ballistics may have an interest in this experiment. Peers interested especially in the study of energy and momentum would have the most interest in this data.

While analyzing the data found from a ballistic pendulum experiment, the Law of Conservation of Mechanical Energy and the Law of Conservation of Momentum both need to be utilized. The involvement of both of these physical laws struck the group's immediate interest because of the pertinence to what had been being learned in class. The two previous laws were used in concert with a background with springs, Hooke's Law, and inelastic collisions. This background knowledge is essential in understanding the experiment, resulting calculations, and analysis. The main objective of this experiment was to determine the initial velocity of a ball shot from a spring loaded gun, into a receptacle which traveled up a ramp. As well as finding the initial velocity of the ball we also wanted to determine the spring constant of the spring used in the spring loaded gun. Lastly, we wanted to analyze what speed would theoretically be reached if no friction forces acted on the apparatus and what percent was lost to friction.

Law of Conservation of Mechanical Energy:

Emech = Ð"K + Ð"U

Kf + Uf = Ki + Ui

If no dissipative forces are involved in the process then this equation holds true and mechanical

energy is conserved.

Law of Conservation of Momentum:

The total momentum of an isolated system is constant thus deriving the above equation. In the case of a perfectly inelastic collision (such as in this experiment) momentum is conserved.

Hooke's Law:

(Fsp) s = -kÐ"s k is the spring constant and Ð"s= s-se is displacement from equilibrium

The ballistic pendulum apparatus that was used to perform the experiment had three main parts. The first part contained a spring loaded gun which a round, metal ball was loaded. Next, a relatively massless rod (movement arm) was hung from a stable point. At the end of the massless rod contained a receptacle in which the ball was shot into. This portion of the apparatus is where the inelastic collision was completed. After the ball was in the receptacle a notched metal portion finalized the apparatus in order to catch and gauge the final height reached by the ball/receptacle.

(Original Picture taken from http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html/)

Method:

Before loading the gun, the length of the spring was measured. Next, a metal ball was placed onto the end of a spring loaded gun. The gun was cocked by pushing on the ball until the spring was compressed far enough in order for an arm to engage and prevent the spring from uncoiling. The spring was measured again in order to obtain a measurement for how far the spring was compressed. The movement arm was steadied, angled correctly in order to receive the ball. When the trigger was released, the ball was shot into the ball receptacle at the end of the movement arm. The arm then swung up to an angled catcher with ridges. The receptacle catches along the ridges and were able to measure how far up the ramp the ball and receptacle system traveled. The change in vertical height was measured by first marking the middle of the ball up on the ramp. The ball was then removed from the receptacle and placed back on the gun. We then measured from the previous mark to the center of the ball when it is on the end of the gun. This data was then used to determine the velocity of the ball, movement arm system along with the initial velocity of the ball by incorporating the Law of Conservation of Mechanical Energy and Law of Conservation of Momentum. Three trials were performed in order to gain an accurate average to base our data. As the ball was shot, we had to make sure it was aimed towards a wall to ensure that if the ball missed the receptacle for any reason that it would not strike another student. Also, the spring was fairly strong and we had to be careful of not pinching our fingers as we attempted to adjust the compression and while loading the gun.

To obtain a spring constant we had to remove the spring from the ballistic pendulum set up. We measured the resting length of the spring, without weight on it. Next, we obtained a 1 kg mass and hung it from the bottom of the spring. The spring was held vertically with the mass attached and a measurement of how far the spring had stretched was obtained. The change in length, along with the force on the spring was used in addition to Hooke's Law, in order to determine the spring constant.

Data and Calculations:

Mass of Ball = 68g = .068 kg

Mass of Movement Arm = 270g = .270 kg

Total Mass of the System (Ball + Movement Arm) = .338 kg

Trial Spring Length (cm) Compressed Spring Length (cm) Compression (cm) Ð" y up the ramp (mm) Ð" Height (cm)

1 8.0 3.6 4.4 15.0 6.0

2 8.0 3.6 4.4 15.0 6.0

3 8.0 3.6 4.4 15.0 6.0

Average in Meters (m) 0.080 0.036 0.044 .0150

0.060