THE BALLISTIC PENDULUM AND THE LAW OF CONSERVATION OF ENERGY Objectives 1) To study the laws of conservation of energy, conservation of momentum, and the elements of projectile motion using the ballistic pendulum. ) To calculate the initial velocity of the Projectile Launcher using the basic principles of projectile motion. Introduction The momentum of an object is the product of the mass and the velocity. The principle of conservation of momentum may be derived from Newton's second law of motion. Simply stated this principle is, If no external forces are acting on a system containing several bodies, then the momentum of the system remains constant. This experiment will be applying this principle to a collision, using the ballistic pendulum. The ballistic pendulum consists of a gun which fires a ball into the pendulum's bob. The initial velocity of the ball is determined in terms of the masses of the ball and the bob and the height to which the bob rises after impact. The velocity also may be obtained by firing the ball horizontally and allowing it to fall freely toward the earth. Using the vertical distance of fall and the range traveled from the projectile problem the velocity is determined. In a collision between two objects the momentum before the impact is equal to the momentum after the impact. Applying this conservation law two the ballistic pendulum, the momentum of the projectile is equal to the momentum of the pendulum and the bob after impact. mv ( m M ) V (1) where m is the mass of the ball, v is the initial velocity of the ball before impact, M, is the mass of the pendulum, and V is the velocity of the pendulum and the ball immediately after impact. The type of collision between the ball and the pendulum prohibits us from using the conservation of energy before and after the collision. Despite this shortcoming, there is another area of the overall experiment that allows us to use the conservation of energy. The energy of the pendulum and ball just after impact is all kinetic energy. It is this energy that is transformed into potential energy when the pendulum and ball swing up and come to rest with its pawl engaged with a tooth on the upper rack. When this is put into an equation we obtain K. E. P. E. (a) initial final 1 ( m M) V ( m M) gh (b) Using equation (b) the velocity, V of the pendulum and ball after impact can be calculated. This velocity is then substituted into equation (1) to obtain the initial velocity of the projectile. The Ballistic Pendulum 1
A second method may be used to calculate the velocity, v of the projectile. This is performed by shooting the ball out horizontally from a height, s above the floor. This is illustrated in the figure below. By measuring the horizontal range of flight, x, and the vertical distance, y, fallen, the initial velocity of the projectile, v, can be found from v y x t 1 gt Procedure: Method I 1) Record the mass of the pendulum arm and the ball. You can measure the mass of the ball on the scales in the lab. The mass of the arm is provided on the board. ) Place a stainless steel ball in the spring gun and ready it for firing by using the ramrod. Ready the pendulum by removing it from the latched position and allow it to hang freely. Set the angle indicator to 0 o. If it is not at 0 o, estimate its starting angle very carefully. When the pendulum is at rest, pull the trigger, thereby firing the ball into the pendulum bob. This will cause the pendulum with the ball inside it to swing up along the rack where it will be caught at its highest point. Record the height, (h ) reached by the ball and pendulum when they come to rest in this raised position. To remove the ball from the pendulum, press the releasing trigger. 3) Repeat step ) a total of five times and record your results. 4) From the data of procedures ) and 3), compute the average value of the position of the pendulum in the raised position and record the answer. 5) With the pendulum hanging in its lowest position, measure the vertical distance of the ballistic pendulum (h 1 ) from the same point used in measuring (h ). The Ballistic Pendulum
Procedure: Method II 6) Set the apparatus near one edge of a table. Level the base by placing a few sheets of paper under the legs, if necessary. Clamp the base to the table. 7) Carefully, convert the ballistic pendulum into the projectile launcher. Prepare the projectile launcher by inserting the ball into the device to the same amount of compression used in Method I. 8) Shoot the ball horizontally in free flight. One observer should note carefully the position where the ball strikes the floor and another person should catch the ball before it strikes the wall or the furniture. Now, place a piece of paper on the floor where the ball will surely strike it, cover with carbon paper and fasten securely to the floor with tape. 9) Fire the ball five times. 10) Measure the horizontal distance, x, for each shot. See figure on the previous page for an illustration. Use a plumb bob to locate the point on the floor directly below the ball as it leaves the gun. 11) Measure the vertical fall of the ball, the vertical distance, y. This distance will be from the point projected on the floor to the bottom of the ball when it is in the spring gun. See the illustration for details. The Ballistic Pendulum 3
Calculations: Method I 1) From the data collected in procedures 1) thru 5), and recorded in the Data Table for Method 1, calculate the average angle, avg, and record your answer in Data Table for Method I. ) Calculate h using the relationship h = L(1 - cos ). h, is dependent upon the pendulum length, L, and the average angle, avg. 3) Compute the velocity of the pendulum and ball V just after the collision using equation b). Record your answer in Data Table for Method I. 4) Calculate the initial velocity of the ball, v, using equation 1) and the velocity, V, of the pendulum and ball calculated in step ) above. Record your answer in Data Table for Method I. Calculations: Method II 5) Calculate the average horizontal distance, x, for the five shots fired and measured in procedures 5) and 6). Record your answer in Data Table for Method II. 6) Using the measured vertical distance, y, and the acceleration of gravity in equation 4), calculate the time of flight for the ball. Record your answer in Data Table for Method II. 7) Calculate the initial velocity of the ball using equation 3) and the time of flight just calculated in step 6). Record your answer in Data Table for Method II. Comparing Results 8) Compare the values of the initial velocity of the ball obtained from both Method I and Method II. Express this comparison as a percent difference and record your answer in the Data Table. 4 The Ballistic Pendulum
Questions 1) What is the kinetic energy of the ball before impact? ) What is the kinetic energy of the pendulum and ball after impact? 3) What percent of the energy of the ball was transferred to the combination of the pendulum and ball? 4) State whether the collision in this experiment was an elastic or inelastic collision. Explain your answer. 5) If, in the free flight determination of the velocity, Method II, the floor were not level introducing an error of 1 cm in the height y, what percent difference would be introduced into the initial velocity? 6) An error in the horizontal distance, x, could also be made when the initial velocity is determined by the free flight method. Calculate the percent difference that would be introduced into the initial velocity, if a 5 cm error was made in the horizontal distance measurement, x. The Ballistic Pendulum 5
Names Data Table for Method I Mass of the pendulum bob Mass of the ball kg kg Angle of ballistic pendulum at highest position, : Average angle of highest position, avg Length of pendulum arm, L: Vertical distance, h, traveled by pendulum Calculated velocity of the pendulum and ball just after the collision (V) Velocity of the ball before the collision (v) /s /s Data Table for Method II Horizontal Distance, x, of the projectile m m m m m Average Horizontal Distance, x: Vertical distance of fall, y Initial velocity of the ball calculated using equation 3) and 4) /s Comparison of ball s initial velocity Difference between Method I and Method II value for velocity, v /s Percent difference of initial velocity % 6 The Ballistic Pendulum