Physics 30 - Ballistic Pendulum Lab 2010, Science Kit All Rights Reserved

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1 BACKGROUND Energy The maximum height achieved by the pendulum on the Ballistic Pendulum apparatus can be determined by using the angle it achieved. Figure S1 shows the pendulum in two different positions, one at its lowest point and one at its highest point. In both cases, the length (L) of the pendulum is the same but the height (h) above the lowest point changed. By applying the trigonometric rules for cosine, we can derive the equation for the maximum height the pendulum achieved based only on the angle and the length of the pendulum arm. S1

2 When the spring launcher is fired, the stored potential energy of the spring is released into the steel ball as kinetic energy. When the ball collides with and sticks in the pendulum catch, its kinetic energy is transformed into gravitational potential energy as the pendulum reaches its highest point. According to the law of conservation of energy, the kinetic energy the moment after impact, at the lowest part of the pendulum's swing, is equal to the gravitational potential energy at its highest point. This is represented as: As a result of the impact, the pendulum containing the projectile swings above its lowest point through a vertical distance (h). Knowing this distance, it is possible to determine initial velocity of the ball-andpendulum system the moment after the impact. Solving the previous equation for v i yields: Once the maximum height (h) is determined, the equation v i = 2gh can be used to find the velocity of the pendulum-and-ball system at the lowest point. S2

3 Momentum The momentum (p) of a body is defined as the product of the mass (m) of the body by its velocity (v). This can be expressed as p = mv. When two objects collide, there is an equal and opposite reaction force. This reaction force affects the velocity of each of the objects in different ways. Imagine an object of mass M moving with a velocity v 1 when it strikes a stationary object of mass m. When the two objects collide, there is an equal and opposite force exerted on each object, and, according to Newton's second law (F = ma), an acceleration as well. This acceleration changes the velocity of the object and, therefore, its momentum. Because the change in velocity will be unique for each object, we can represent the change in velocity for mass M as v 1 and the change for mass m as v 2. When two objects collide, the total momentum of the system must be conserved. This can be represented as: p before = p after When the two objects remain separate from each other after the collision, such as in playing pool, the collision is referred to as an elastic collision. The cue ball collides with a second ball and they both go their separate ways, and each with their own velocity. The momentum of an elastic collision can be represented as: In the case above, the velocity of m is zero before the collision and the velocity of M is zero after the collision. Therefore, the equation can be written in a more simplified version: When two objects collide and are locked together after the collision, as in the ballistic pendulum, they have the same final velocity. This type of collision is referred to as inelastic. S3

4 In the ballistic pendulum apparatus, the velocity of the pendulum before impact is zero, and, hence, its momentum before impact is zero. The momentum of the system can be written in a simplified form as: If the initial velocity of the launched steel ball is to be determined (V i ), the equation can be rearranged into: Determining Initial Horizontal Velocity The actual velocity of the launched steel ball can be determined either by analyzing its horizontal projectile motion. In this method, the pendulum must be held out of the way so the ball can fly freely. Projectile Motion In projectile motion, the horizontal component of the velocity always remains constant; it is not affected by the downward pull of gravity. Also, the vertical component of the motion is unaffected by its horizontal flight; the body falls vertically, as does a freely-falling object having no horizontal motion. The actual motion of such a projectile is the combination of its horizontal constant velocity and its downward uniformly-accelerated velocity. During the time interval (t) required for the projectile to reach the floor, it will have moved horizontally through a distance of: x = vt During the same interval, because of the acceleration due to gravity (g), it will fall a distance of: y = gt2 2 By eliminating t between these two equations and substituting the measured values of x and y, the initial velocity may be found by the equations and steps below. t = x v t = 2y g x v = 2y g v = x2 g 2y S4

5 ACTIVITY 1: INTRODUCTION TO CONSERVATION OF MOMENTUM Materials Needed: Ballistic Pendulum Masking Tape Steel Ball Prying rod Slotted Mass Procedure: 1. Set the Ballistic Pendulum so that it will launch the steel ball into the catch. 2. Set the launch mechanism at the first slot and fire. Note the maximum angle the pendulum achieves and record it in Data Table 1. Use the prying rod to remove the steel ball. 3. Repeat step 2 four more times to ensure results are consistent. 4. Tape a slotted mass to the bottom of the pendulum catch. Now that the pendulum catch has more mass, predict what the maximum angle will be and record it in Data Table Set the launch mechanism at the first slot and fire. Note the maximum angle the pendulum achieves and record it in Data Table 1. Repeat four more times to ensure results are consistent. 6. With the mass still attached, set the launch mechanism at the second slot. Predict what the maximum angle will be and record it in Data Table Fire once ready and record the results. Repeat four more times to ensure results are consistent. 8. With the mass still attached, set the launch mechanism at the third slot. Predict what the maximum angle will be and record it in Data Table Fire once ready and record the results. Repeat four more times to ensure results are consistent. 10. Calculate the average maximum recorded angle for each. (2 marks allocated in rubric) ACTIVITY 1 Analysis Questions: (answer these in your analysis section) 1. What general rule can you make about the mass of the pendulum catch and the distance it will rise? /2 (allocated in rubric) 2. How does the launching position affect the distance the pendulum will rise? Explain it in terms of energy. /2 (allocated in rubric) S5

6 ACTIVITY 2: CONSERVATION OF MOMENTUM via PROJECTILE MOTION Materials Needed: Ballistic Pendulum Prying rod Steel Ball Block of wood 30cm Ruler Carbon paper Plain paper String Procedure: 1. Tape a 10cmx10cm square or carbon paper to a block of wood with the carbon side facing up. 2. Fasten a piece of paper on top of the carbon paper so that the bottom edge of the paper will rest squarely on the surface of the tabletop. 3. Place the target against the edge of the ballistic pendulum apparatus so that it is lined up with the launch mechanism. 4. Using string, tie the pendulum catch out of the way (up and to the right) of the path of the projectile. 5. Set the launch mechanism at the first slot and fire. Repeat this 4 more times. 6. Measure the initial height of the steel ball from table top to the center of the firing plunger. Record the value in Data Table 2A next to y i (m). 7. Remove the piece of plain paper from the target and measure the distance from the bottom of the piece of paper up to the centre of each black dot. Find the average height and record the value in Data Table 2A next to y f (m). 8. Measure the range of the projectile and record the value in Data Table 2A next to x (m). 9. Measure the mass of the steel ball. Record the value in Data Table 2B next to m (g). Note: The mass of the pendulum and catch (M) is provided for you, recorded as g 10. Measure the length of the pendulum arm from the pivot point to the center of the pendulum catch. Record the value in Data Table 2B next to L (m). 11. Record the average maximum recorded angle of No mass + fired at position 1 from Data Table 1 into Data Table 2B and place it next to ( ). 12. Once all your data is collected, put away all the lab equipment. S6

7 ACTIVITY 2 Analysis Questions: (answer these in your analysis section) 1. Calculate the maximum height, h (m), the pendulum achieved. Refer to page S1 as needed. Record your results in Table 2B. /2 (allocated in rubric) 2. Use the equation for conservation of energy to determine the velocity of the steel ball and pendulum catch after the collision, v f (m/s), at the lowest part of the swing. Refer to pages S1 and S2 as needed. Record your results in Table 2B. /2 (allocated in rubric) 3. Use the equation for conservation of momentum to determine the initial velocity of the steel ball, v i (m/s), from the launcher. Refer to pages S3 and S4 as needed. Record your results in Table 2B. /2 (allocated in rubric) 4. Determine the free fall distance the ball fell during projectile motion y. Record your results in Table 2A. 5. Using the projectile-motion method, determine the velocity of the steel ball at the moment it was launched. Refer to page S4 as needed. Record your results in Table 2A. /2 (allocated in rubric) 6. Determine the percent difference between the predicted steel ball velocity based on energy transfer and conservation of momentum, Table 2B, and the predicted steel ball velocity based on projectile motion, Table 2A. /2 (allocated in rubric) 7. What do your results from part 6 inform you about the possibilities of error in this experiment? /2 (allocated in rubric) Conclusions Summarize activities 1 and 2 and state whether or not your results support or refute any current theories of motion or energy transfer. Provide any relevant applications and where future improvements to lab, or recommendations, could be made. Point out at least 3 sources of error with a reasonable explanation as to how they affected the data. Use the lab write up guidelines provided for you in your first lab. S7

8 Problem: Clearly defined purpose with a focused topic of study /2 Theory: Clearly states the physics background of lab (Equations explained) /5 Materials: All necessary materials are listed. /2 Procedure: Specific and reproducible. /5 Observations: Any details witnessed with the senses and physical measurements made. /4 Measurements are to be placed in neat and organized charts. /6 Analysis: 1. Questions answered (well constructed sentences, demonstrates understanding) /6 Conclusions: 2. Example calculations (clearly demonstrates logical reasoning) /12 Problem is answered (supported or refuted) based on data. Relevant applications are made. Future improvements to lab are listed. At least 3 sources of error are identified with a reasonable explanation as to how they affected the data. Attach this to your lab. /10 /6 S8