Modeling Projectile Motion

Introduction

Build a simple model to illustrate the parabolic shape of projectile motion. The assembled “Projectile Motion Model” can be used to estimate the speed of a projectile by “dialing in” its path.

Concepts

• Acceleration due to gravity
• Parabolic motion
• Newton’s laws of motion
• Projectile motion

Materials

Safety Precautions

The materials in this demonstration are considered safe. Do not aim a projectile launcher in the direction of other people. Wear safety glasses when using a projectile launcher. Please follow all normal laboratory safety guidelines.

Prelab Preparation

Assembly 

1. Attach the metal clips onto the meter stick at the 0-cm, 10-cm, 20-cm, etc., marks as shown in Figure 1.
   {13540_Preparation_Figure_1}
2. Cut 9 pieces of string to the following lengths: 6 cm, 10 cm, 15 cm, 20 cm, 30 cm, 55 cm, 70 cm, 90 cm and 110 cm. Use a tape measure, meter stick or metric ruler to measure the length of each piece of string.
3. Securely tie a fishing sinker to one end of each piece of string using a double-overhand knot. Tie the fishing sinker as close to the end of the string as possible.
4. Thread the shortest string through one of the wire arms of the metal clip at the 10-cm mark on the meter stick (see Figure 2).
   {13540_Preparation_Figure_2}
5. Thread the free end of the string through a button clamp to secure it to the metal clip (see Figure 2).
6. Repeat steps 4 and 5 for the eight remaining pieces of string and their attached fishing sinkers. Use a longer string length for each consecutive position (see Figure 3).
   {13540_Preparation_Figure_3}
7. Refer to Table 1 for the proper string lengths for each position. Adjust the string length by pulling the string through the button clamp until the appropriate hanging length is achieved. Measure between the end of the wire arm on the metal clip and the knot on the fishing sinker. Use a tape measure, meter stick or ruler to measure the length of each piece of string. Hold the Projectile Motion Model high enough so that all the fishing sinkers hang freely. (This may require a second pair of hands to hold the Projectile Motion Model.)
   {13540_Preparation_Table_1_String lengths}
8. Once each position has the proper string length, any excess string protruding from the button clamp may be clipped off, if desired. Leave about 1 cm of string at the top to make sure the button clamp fastens to enough string to keep the hanging fishing sinkers secure.
9. The assembled Projectile Motion Model can be held in front of the class to show the parabolic shape of a projectile. Alternatively, the metal clip at the zero mark can be slipped onto the included support rod and clamped to a support stand. The end of the meter stick can then be raised to show projectile motion at different angles (see Figure 4).
   {13540_Preparation_Figure_4_End view}

Procedure

Modeling Projectile Motion

1. Show students the assembled Projectile Motion Model and discuss the concepts of acceleration due to gravity, constant speed, projectile motion, constant acceleration, Newton’s laws of motion, etc., as appropriate for your level of classroom instruction.
2. Explain to students that each string position along the meter stick represents a 0.05-second increment of time. In other words, the fishing sinker on the first string represents the position (x, y) of a projectile after 0.05 seconds. The fishing sinker on the second string represents the position of a projectile after 0.10 seconds, and so forth. So, when the metal clips are attached at 10-cm intervals, the entire length of the meter stick (100 cm) is traversed in 0.5 seconds. The last fishing sinker represents the distance a projectile falls under a constant acceleration of 981 cm/s² (g) in 0.45 seconds.
3. Students may recognize the shape of the model as that of a parabola. Explain that the parabolic shape is the result of the “time-squared portion” of acceleration. The distance a falling object drops is related to time squared rather than just time (which would produce a linear model).
4. Show different projectile motion angles: Raise or lower the angle of the Projectile Motion Model to change the shape of the parabola. Remember to keep the zero position at the same height.
5. Position the angle of the model at approximately 45° to the horizontal to show students the angle at which a projectile will launch the farthest distance.
6. Relate the position of the last fishing sinker to lower and higher angles in order to illustrate why 45° is the best angle.

   Note: At slightly smaller angles, the horizontal position of the last sinker does not change much compared to its position at 45°, but its height is lower. Hence, the projectile will not travel as high compared to a projectile fired at a 45° angle and therefore will not travel as far because it will not be in the air as long. At an angle slightly above 45°, the last sinker is slightly higher, but it has not traveled as far horizontally as the 45° path. This illustrates that the projectile will travel higher compared to one fired at 45°, but it will fly a shorter distance away from the origin because it has traveled a shorter horizontal distance in the same amount of time.

1. Set up an inclined plane to roll a marble or ball bearing off the end of a tabletop. (Optional) Set up a projectile launcher to fire horizontally. A smooth transition between the bottom of the inclined plane and the tabletop is necessary so the ball will not bounce significantly as it rolls off the end of the table. Bent thin metal sheet, or thin cardboard may be used as the transition material. Use tape to secure it to the bottom of the inclined plane and tabletop (see Figure 5).
   {13540_Procedure_Figure_5}
2. Position the Projectile Motion Model to the side of the inclined plane so that it will not interfere with the motion of the projectile (ball) (see Figure 5). Hold the Projectile Motion Model parallel to the ground (or have a student hold it). String can also be used to tie the Projectile Motion Model and hold it parallel to the ground. However, it is often easier to have a student hold it.
3. Line up the zero mark on the meter stick with the origin of projectile motion (the end of the tabletop) (see Figure 5). 
4. Starting at a medium height, roll the marble down the inclined plane.
5. Students should observe the path of the projectile and compare it to the shape produced by the Projectile Motion Model. If the path is not long enough, roll the ball bearing from a higher starting position.
6. Encourage students to suggest how the spacing between the metal clips should be adjusted in order to produce the proper shape of the projectile. If necessary, remove string positions 7, 8 and 9 from the meter stick to allow for the increased spacing between the remaining clips. Remember: The spacing between each clip must be the same in order to model a constant horizontal speed.
7. Launch the projectile several times using the same settings (initial height) until the parabolic shape is correctly displayed.
8. Once the parabolic shape of the Projectile Motion Model is properly aligned with the actual path of the projectile, the speed of the projectile can be determined. Measure the distance between each metal clip (it should be approximately the same between each pair of clips). Divide the distance by 0.05 seconds to calculate the horizontal speed of the projectile.

Student Worksheet PDF

13540_Student1.pdf

Teacher Tips

• This kit contains enough materials to assemble one Projectile Motion Model. The model can be reused indefinitely.
• Copy the Projectile Motion Model Worksheet for each student, if appropriate for your classroom setting. Students may complete the worksheet during the demonstration to improve their understanding of the concepts being taught.
• Figure 6 shows a general illustration of the setup using a projectile launcher. For more advanced students, “dial in” the initial speed of a projectile shot at an angle.
  {13540_Tips_Figure_6}
• Fire the projectile at a low setting (or roll a marble down an inclined plane from a medium height). The Projectile Motion Model has a limited “dialing in” range so make sure the projectile does not travel too fast. Fewer than five hanging strings may make it difficult for students to see the shape of the parabolic motion.
• A small amount of error in the parabolic shape will occur if the end of the meter stick is not lined up with the end of the tabletop. Usually this is negligible.
• A C-clamp or Quick-Grip^® style clamps can be used to secure the meter stick to the end of a tabletop. The meter stick and materials are light enough to be supported even if clamped at the very end.
• Before using a projectile launcher for the demonstration, practice firing it to estimate the distance the ball will travel as well as the shape of the parabola. Use a launch setting that will fire a projectile within the shape limits of the Projectile Motion Model. The maximum speed for an effective display of the parabolic motion is approximately 500 cm/s (11 miles/hr) in the horizontal direction. This model would have four hanging strings spaced 25 cm apart.
• The string positions can be moved anywhere along the meter stick to represent different trajectories. However, in order to represent constant horizontal motion, each string position must be spaced the same distance apart. For example, when each position is at 10-cm increments, the model would represent an object traveling with a horizontal speed of approximately 200 cm/s since each interval represents 0.05 seconds. To model an object traveling at 400 cm/s, each string position will need to be 20 cm apart. Only the first five string segments would be clipped to the meter stick when modeling an object traveling at 400 cm/s.
• Use potential (mgh and ½Kx²) and kinetic energy (½ mv² and ½Iω²) equations to calculate the theoretical value of the speed of the projectile and compare it to the “dialed in” speed.
• Additional kits that help teach about projectile motion, acceleration due to gravity, and Newton’s laws of motion include the Guinea and Feather Tube (Catalog No. AP4670), Shoot the Monkey Demonstration Kit (Catalog No. AP6439), Rubber Band Cannon (Catalog No. AP6624), Second Law of Motion Apparatus (Catalog No. AP4737), Trajectory Apparatus (Catalog No. AP9231), Inclined Plane, Economy Choice (Catalog No. AP4535) and the Projectile Launcher (Catalog No. AP5696). These kits are available at Flinn Scientific, Inc.

Further Extensions

Demonstrate the properties of periodic pendulum motion using the Projectile Motion Model. No changes need to be made to the lengths of string. Simply oscillate all fishing sinkers at the same time and have students observe their motion. Each pendulum will oscillate with a different frequency, even though each mass is the same. This is an excellent demonstration to show students that a pendulum’s frequency (and period) depends only on the length of the pendulum and not on the mass of the pendulum bob.

Answers to Questions

1. What shape do all projectiles follow?

   All projectiles follow the shape of a parabola (in the absence of air resistance or other outside forces).

2. Starting from rest, how far will an object fall in one second?

   490.5 cm

3. How far does an object with a constant horizontal speed fall in one second?

   490.5 cm

4. What is the best angle for launching a projectile in order to achieve the greatest distance?

   45°

5. What is the best angle for launching a projectile in order to achieve the greatest height?

   90°

6. (Optional) Show the calculations needed to determine the horizontal speed of a projectile using the “dial in” method.

   Metal clip separation = 15 cm
   15 cm/0.05 s = 300 cm/s
   (Student answers may vary.)