Shoot the Monkey

Introduction

A very clever monkey named Mittens has escaped from a zoo. Mittens climbs up a tree to escape the pursuing zookeeper. The only way the zookeeper can get her down now is by hitting her with a stun dart. However, Mittens is wise to the zookeeper’s plan. Not wanting to be hit with the dart, Mittens believes that if she lets go of the tree branch at the same time the zookeeper shoots the dart, the dart will sail over her head as she falls safely to the ground to flee to another tree. The zookeeper knows that Mittens is a very clever monkey. He has a feeling that Mittens will let go of the tree branch as soon as the dart is fired. Where should the zookeeper aim the dart in order to hit the monkey as she falls? Fortunately, the zookeeper knows something that Mittens doesn’t—the zookeeper knows a little something about physics.

Concepts

• Acceleration due to gravity
• Projectile motion
• Falling objects

Materials

Safety Precautions

Wear safety glasses when performing this demonstration. The dart is tethered to the backboard so it will have limited range if it misses the backboard. However, the dart tip is dangerous. Do not aim the dart at anyone and make sure no one stands behind the backboard during the demonstration. Observers should be at least ten feet away from the demonstration apparatus when the dart is released. Follow all laboratory safety guidelines.

Disposal

The materials for this kit should be saved for future demonstrations. Save the plastic safety guard that is placed on the tip of the dart during shipment. Replace the safety guard on the tip of the dart during storage. If this safety guard is misplaced, place a drinking straw over the dart tip and tape the straw to the dart. Make sure the drinking straw extends beyond the dart tip by at least a centimeter to prevent the tip from injuring anyone.

Prelab Preparation

Backboard Assembly

1. Obtain the wood backboard, wood base, three angle-brackets, and six flat-head screws.
2. Connect the predrilled backboard and base together with the three angle-brackets (two on the back and one in front) and six screws according to Figure 1. Use a Philips-head screwdriver to tighten the screws.
   {13887_Preparation_Figure_1_Backboard assembly diagram (not to scale)}
3. Use a Philips-head screwdriver to attach the metal handle to the top of the backboard with the two machine screws. Refer to Figure 1.
4. Wrap the thick rubber band around the backboard handle so that it stretches around the ends (see Figure 2a).
   {13887_Preparation_Figure_2}
5. Twist the rubber band three or four times with your fingers so that the top band is twisted toward you and the bottom band is twisted away from you (when looking at the front of the handle) (see Figure 2b).
6. While holding the rubber band twisted, insert the small end of the 3½-inch nail between the twisted bands in the middle of the handle (see Figure 2c). Do not poke the nail through the rubber.
7. Release the nail. It should spring back so that the head of the nail is pressed against the backboard and the point end is extended out (see Figure 1). If it rotates in the opposite direction (the head is extended away from the backboard and the point is pressed against the backboard) remove the nail, twist the rubber band in the opposite direction three or four times, and then reinsert the nail according to Figure 2c.
8. Adjust the nail so that about half is above the handle and half is below the handle.

Monkey Target Assembly

1. Photocopy the monkey figure sheet.
2. Cut out one monkey figure from the photocopied sheet with scissors.
3. Position the monkey figure cutout close to one corner of the foam sheet and then use scissors to cut out the foam sheet in the shape of the monkey figure cutout. The foam sheet is large enough to get two monkey-shaped cutouts.
4. Tape the monkey cutout to the foam sheet with transparent tape. Tape all the edges of the monkey cutout completely so that no loose flaps are present. (Loose flaps will increase the drag on the target as it falls.)
5. Tape the three large washers to the back of the monkey target in a triangular pattern near the bottom of the target (see Figure 3). Make sure the monkey target is balanced across the midline so that it falls straight. The washers serve a dual purpose: (1) to give the target extra weight so the foam will fall without being as easily affected by air drag, and (2) to give smooth contact points between the target and the backboard so frictional drag against the backboard is minimal.
   {13887_Preparation_Figure_3}

Dart Launcher Setup

1. Refer to Figure 4 for a diagram of the dart launcher setup.
   {13887_Preparation_Figure_4}
2. Use a C-clamp or heavy books to secure the base of the backboard assembly to a tabletop or on the floor.
3. Measure and cut one 60-cm length of string and two 30-cm lengths of string.
4. Tie a loop knot (see Assembly Tips Figure 6) at one end of 60-cm string.
5. Secure the 60-cm string to the backboard handle by making a “looping” knot (see Assembly Tips Figure 8) with the loop-knot end of the string. The “looping” knot should be in the center of the handle. 
6. Tie the free end of the string secured to the backboard handle to one end of the elastic band. Make sure the knot is as close to the metal barb as possible. The metal barb will prevent the string from slipping off the elastic band.
7. Tie one of the 30-cm strings to the free end of the elastic band.
8. At the free end of the string just tied to the elastic band, tie a loop knot.
9. Tie the end of this string to the front of the dart using a “looping” knot (see Assembly Tips Figure 7). Secure the string to the metal part of the thicker dart shaft; do not tie on the dart point. Pull the string tight so the dart is bound tightly to the string. Transparent tape can be used to secure the string on the dart if necessary.
10. Tie a loop knot at the end of the second 30-cm string.
11. Secure this string to the back of the dart using a “looping” knot (see Assembly Tips Figure 7). The loose end of this string will be used to pull the dart and release it. The dart fins will prevent the string from slipping off the dart.
12. Practice pulling the dart away from the backboard and releasing it. Hold onto the loose string tied to the back of the dart and pull the dart back so that it is parallel to the ground. Aim the dart at the backboard handle where the dart cord is tied. Release the dart horizontally and observe where it hits the backboard.
13. Determine the appropriate stretch distance for the dart so that the dart hits the backboard at or below its midpoint. It will take some practice to get a general sense of how much elastic stretch is needed to launch the dart appropriately. Decreasing the stretch distance of the elastic band will launch the dart with less velocity and it will hit the backboard at a lower position.
14. Once the stretch distance is determined, obtain the spool of string.
15. Tie the free end of the spool of string to the top of the nail inserted into the twisted rubber band (see Figure 5). The knot should be just below the nail head. Use transparent tape to secure the string to the nail and prevent it from slipping if necessary.
    {13887_Preparation_Figure_5}
16. Unravel the string from the spool until the release string is as long as the appropriately stretched dart cord. Cut the string so that it is about 30 cm longer than the properly stretched dart cord.
17. Pull on the string tied to the nail (the release mechanism) so that the nail head rotates out and the nail point presses against the backboard. Do not pull hard enough to pull the nail out of the rubber band.
18. At the same time, pull the dart cord back to the appropriate release distance determined in step 13.
19. When the release string and dart cord are taut and at the appropriate distance away from the backboard, and the nail point is pressed against the backboard, pinch the release string and dart cord together with your fingers at a point 10 to 15 cm behind the dart.
20. While maintaining the relative positions of the ends of the two pieces of string with your fingers, tie the two pieces of string together. This knot will act as the holding point. When this knot is pulled horizontally until both the release string and dart cord are taut, the elastic band will be at the appropriate release distance, and the release string will pull the release mechanism tight enough to hold the monkey target in position.
21. Practice pulling, aiming and releasing the dart a few times. If adjustments need to be made, untie the holding-point knot and repeat steps 17–20.
22. Practice shooting at the monkey target by placing the target between the handle and the backboard. Position the middle of the target so that it is in line with the “looping” knot that is tied to the middle of the handle (see Figure 5).
23. Pull on the release string in order to rotate the nail to hold the target securely in place. Do not pull too hard on the release string and pull the nail out of the rubber band twists. Pull just hard enough to keep the target in position. This will take some practice.
24. Maintain the tension on the release string as you slowly move backward away from the backboard. Hold the release string tight enough to keep the tension, but loose enough to let it slide through your fingers as you walk backwards. The dart cord is tethered to the release string, so as you move away from the backboard, the dart should be pulled up and towards you as the elastic band begins to stretch. At first, it may take two people to practice. One person to hold the target, and one person to pull the dart until the release mechanism and dart are in the appropriate position. Caution: Make sure the target holder moves at least 10 feet away from the backboard before the dart is released.
25. Once the dart is in the appropriate position, make sure it is aimed horizontally and at the monkey target.
26. Release the dart and observe the dart strike the target in midair. The “expected” dart impact on the target should be near the point on the target that was lined up with the “looping” knot tied to the handle before it was released. Very small timing errors can cause the dart to miss the “expected” region. See Assembly Tips for recommended adjustments.
27. Practice the dart release/target drop procedure several times before demonstrating in front of the class.

Procedure

1. Place the monkey target between the backboard handle and the backboard so that the monkey’s body is lined up with the “looping” knot attached at the middle of the handle (see Figure 5 in the Prelab Preparation section).
2. Pull on the release string in order to rotate the nail to hold the target securely in place.
3. Maintain enough tension on the release string to hold the target in place as you slowly move away from the backboard and pull the dart back to the appropriate distance. It may be helpful to have a student volunteer stand to the side and hold the target as the dart is pulled into position. Caution: Beware of accidentally releasing the dart. Once the dart and target are appropriately positioned, make sure the target holder moves at least 10 feet away from the backboard before the dart is released.
4. Once the dart and target are in the appropriate position, aim the dart so that it will be launched parallel to the ground and directly at the monkey target. The tethered elastic band cord tied to the middle of the backboard handle will act as the guideline to line up the dart with the middle of the target.
5. Release the dart and observe the dart strike the target in midair.
6. Repeat the demonstration as often as necessary.

Student Worksheet PDF

13887_Teacher1.pdf

Teacher Tips

• Enough materials are provided in this kit to perform the demonstration indefinitely.
• Practice, Practice, Practice! Practice launching the dart at the target several times before demonstrating in front of the class. Make sure you are comfortable with the dart launch and target drop procedure.
• The dart position may be adjusted so that it is launched from the “ground” instead of just horizontally at the target. Adjustments to the release distance may be needed in order to keep the release mechanism tight against the target before launching the dart. The tethered elastic band cord will always aim the dart toward the target as long as the target is lined up with the “looping” knot on the backboard handle.
• Make sure to launch the dart in a direction that is parallel to the ground or below. Observers to the side should be able to tell you if the dart is too high (above a parallel-to-the-ground level).
• Go to http://nssdc.gsfc.nasa.gov/planetary/lunar/apollo15info.html to find a QuickTime movie of David Scott dropping a hammer and feather on the moon (accessed February 2012).
• Occasionally, the dart may not stick into the foam target. To better see where the dart hits the target, color the tip of the dart with different water-colored markers before each launch. Different colors can be used for different classes so the same target can be used over and over during the day, and each class will see “their” mark. Make sure to wipe the tip off with a wet paper towel before changing colors.

  Assembly Tips

• The length of string used in this procedure is only one recommendation. Longer launch distances can be achieved, if desired. However, it is more difficult to aim correctly to hit the target at longer distances. Also, the more excess string there is in front of the dart before the launch, the more likely the string may interfere with the flight path of the dart and cause the dart to miss the target. Finally, longer string has a tendency to get tangled up after the launch. It may take a longer time to untangle the string before the next launch.
• Double-wrapping the rubber band around the handle will give the rubber band more “spring” when it is twisted. This may help to release the target faster. It will also hold the nail tighter to prevent it from being pulled out.
• It is best to use “looping” knots or slip knots at the connecting points on the dart to allow for easy disassembly. If the string becomes tangled, the dart can be removed and the string can be more easily unwound.
• Useful Knot Diagrams
  {13887_Tips_Figure_6_Loop knot}
  {13887_Tips_Figure_7}
  {13887_Tips_Figure_8}
  Any knot-tying technique can be used to secure the string to the backboard and dart. The “looping” knot is convenient so that the string can be easily untied from the dart and backboard handle if necessary. “Looping” knots are also convenient to use to tie the string to the elastic band. This way the string can be easily untied from the elastic band if the string gets tangled. Loop knots will need to be tied to both ends of the string secured to the handle and the front of the dart in order to tie “looping” knots to the elastic band.
• The foam sheet may need to be pressed between two heavy books overnight in order to uncurl it after shipping.
• It is best to use transparent tape to secure the washers to the back of the monkey target instead of masking tape. Transparent tape is smoother and will allow the target to slide down the backboard with less resistance.
• There is very little room for error in this demonstration because the time duration of the drop is very short. The target is only about 20 cm long. A target drop delay of about 0.2 second will cause the dart to miss the target. A delay of only 0.1 second will cause a shift by about 5 cm from the expected point of impact (about as long as the monkey’s belly region). It is important that the target drops simultaneously when the dart is released. Make sure the target does not get hung up on the twisted rubber band, the wood backboard or the point of the nail.
  • Move the rubber band and nail release mechanism as close to the end of the handle as possible so that it will not interfere with the dropping target.
  • Do not pull too hard on the release string to cause the nail to pull out of the rubber band or the nail point to dig into the foam. Hold the release string with just enough tension to prevent the target from falling.
  • Do not position the monkey target as high or higher than the head of the nail. When the target is released, the nail will flip around quicker than the target will fall and head of the nail will hit the target and may cause interference with the drop. To position the target higher on the backboard, it will be necessary to shift the position of the nail left or right in the rubber band twist so that the head of the nail will not flip around and hit a piece of the target when it is released.

Discussion

What Galileo (1564–1642) proposed and Newton (1643–1727) verified is that all objects fall toward the Earth at the same increasing rate (in a vacuum). That is, all objects will accelerate toward the Earth equally, regardless of their mass. In a vacuum, where there is no drag friction due to air, a heavy hammer will fall at exactly the same rate as a light feather. (This was demonstrated during the Apollo 15 moon landing when David Scott dropped a hammer and feather at the same time and watched them hit the lunar surface at the same time, proudly announcing that Galileo was correct.) At the surface of the Earth, the acceleration toward the center of the Earth experienced by all objects is measured to be (on average) 9.8 m/s² (32 ft/s²).

Newton also demonstrated that forces can be separated into horizontal and vertical components that are independent of each other. Thus, for a force that pushes a ball up at an angle with respect to the ground, the force is said to have one force component that is vertical and one that is horizontal. Both force components depend on the total force and on the angle of the force with respect to the ground, but they are independent of each other. A vertical force will have no effect on the magnitude of the horizontal force component, and vice versa.

In the Shoot the Monkey demonstration, both the constant acceleration of gravity and the independence of component forces can be observed and studied. The two objects (the dart and monkey target) are released from the same height at the same time. Because they are both acted on by the force of gravity that pulls them toward the Earth with the same acceleration, the dart and target will fall the same distance in a given amount of time. This means that as long as they are released at the same time, they will always be at the same relative height in a given time.

The component forces acting on the dart are a horizontal force from the pull of the stretched elastic band launching it toward the backboard, and a vertical force of gravity pulling down. The target is initially at rest with no vertical or horizontal movement. The target will only be acted on by the pull of gravity. When the dart is released, it is launched horizontally and it begins to fall with a constant acceleration of 9.8 m/s². The target drops at the same moment the dart is released and also begins to fall with an acceleration of 9.8 m/s². The dart and target will always be at the same relative height as they fall, but the distance between them quickly decreases as the dart travels toward the board. No other vertical forces act on the dart or the target besides gravity so each object will continue to fall at the same rate until the horizontally traveling dart hits the target. The monkey target’s path is just a straight vertical line. The dart’s path is not straight—it is in the shape of a parabola.

Optional Advanced Material
The derivation of the kinematics equations for the Shoot the Monkey demonstration follows. This solves for the situation in which the zookeeper is on the ground shooting at a monkey in a tree. Refer to your physics textbook for more information about projectile motion and component forces.

Dart’s Motion
Assume that the elastic band’s force acts over a very short duration of time so that the dart is launched from the ground (y_i = 0) toward the monkey in the tree (y = h) with an initial velocity, v, and at an angle, θ, with respect to the ground. The horizontal and vertical components of the dart’s velocity vector are therefore v_x = v cos θ; v_y = v sin θ, respectively. The time it takes for the dart to travel the horizontal distance d is equal to: t = d/v_x.

In the same time, t = d/v_x, the dart’s vertical position (y_f) will change as a function of the initial vertical velocity component (v_y) and of the constant acceleration due to gravity, –g (the negative sign designates downward acceleration). (Note: sin θ/cos θ = tan θ = h/d.) Monkey’s Motion
The monkey is initially at rest and then only moves in the vertical direction as a result of the force of gravity pulling it down. The monkey’s vertical position after the same flight time of the dart (d/v_x) is shown in the following equations. Equation 6 and Equation 8 are identical, meaning the final height of the dart and monkey target will be the same after the dart has traveled the horizontal distance, d. As long as the zookeeper aims the dart directly at the monkey, and the dart and monkey are released simultaneously, the dart will hit the monkey on the way down.