# Nerf gun

## Executive Summary

Nerf guns are a popular children's toy that comes in a variety of models with different shapes and sizes, different firing mechanisms, and different types of ammunition. For our product study, we chose to focus on three models, The Big Bad Bow, the Rapid Fire 20, and the Buzz Saw. We chose these three products due to their distinct differences in firing mechanisms, appearance and overall feel. For each of these products, we conducted a user study as well as a performance study. During our user study, we found that while users generally enjoyed firing the Nerf guns, they felt that certain aspects of the guns were disappointing and could use improvements to performance. Our performance study confirmed that the maximum firing distance was not very far, with maximum shot distances ranging from 18 to just over 30 feet. The Big Bad Bow left the most room for improvement, being the first choice of users due to its appearance, but also the most disappointing with the shortest firing distance (just over 18 feet). As a result, we chose to further study the Big Bad Bow for our user study and performance study.

Upon dissecting the Big Bad Bow, we found that while the toy contained many parts, all of these parts were either injection molded or purchases hardware. DFMA analysis revealed the product was well designed for manufacture and assembly with all parts being made using one process, and all part being assembled into on side, using common fasteners. FMEA analysis revealed that the gun has a low chance of failure. DFE analysis revealed that much of the environmental impact of the product is due to the energy required for product, leading us to target designing for a less energy intensive manufacturing process. Mechanical analysis of the Big Bad Bow revealed the by increasing the force directly applied to the ammunition, the firing distance can be increased.

## Product Study

We conducted a product study of three Nerf guns, the Big Bad Bow, The Buzz Saw, and the Rapid Fire 20, to determine their functionality, measure their performance, and to gather user feedback and suggestions. From the results of the product study, we decided the the Big Bad Bow provided the most room for improvement. During our user study, users were most attracted to the Big Bad Bow. They often selected this bow first due to its appearance, stating the they would like to test that Nerf gun because it "looks cool." While trying to operate the Big Bad Bow, many users struggled to figure out how to fire, and some grew frustrated with the difficultly of firing the bow. Upon firing the bow, users expressed disappointment in its performance, and stated that they expected it to shoot further or do something more interesting. Because of this marked difference between user expectation and user performance, there is opportunity to produce an improved design that is easier to operate and perform better to increase user satisfaction.

### Functionality

Below are descriptions of the procedures for the designer's intended use of the three blasters. Video Link of all Nerf Blasters Firing at Once

General Description: The Big Bad Bow designed to mimic a compound bow. The ammunition are long foam cylinders launched with a spring loaded air piston. The user pulls back the piston and then pulls on the trigger to release the projectile. The arms of the bow are non-functional.

Step 1: User picks up the fully loaded blaster to hold by both handles.
Step 2: User aims at a target through the scopes.
Step 3: User pulls back the rear-most handle from the rest of the gun till a click is heard or felt.
Step 4: User pulls the trigger which fires the dart and unlocks the bolt.
Step 5: User retracts rearmost handle's sub-assembly.
Step 6: User loads second and third darts from the dart holders subsequently, and then follows steps 2 through 5 to fire darts.
Step 7: User retrieves the darts.
Step 8: User loads darts onto the gun's nozzle and the dart holders.

NOTE: The Big Bad Bow also has an initial assembly procedure for the consumer to perform prior to use. That procedure is straightforward, and not described here.

#### Rapid Fire 20

General Description:The Rapid Fire AS-20 is capable of automatically firing a series of 20 shots with one pull of the trigger in rapid fire mode or 20 individual shots in single shot mode. The firing mechanism and barrel rotation are powered by a compressed air tank that is charged by pumping a handle 25-30 times. The ammunition are 4in long foam cylinders with rubber suction cups on the one end.
Rapid Fire 20 Video

Step 1: User picks up the fully loaded blaster by the yellow handle.
Step 2: User pumps the orange handle 25 to 30 times to fully prime the blaster.
Step 3: User chooses single shot mode or rapid fire mode with the orange dial located on the side of the blaster.
Step 4: User points the blaster in general direction of target(s).
Step 5: User pulls the trigger to fire single darts or a stream of darts.
Step 6: User retrieves darts (20 per clip).
Step 7: User reloads the darts into the barrels of the gun.

#### Buzz Saw

General Description: The Buzz Saw uses the inertia of a flywheel to propel it’s projectiles forward. Angular momentum is accumulated on the ~4 inch plastic flywheel by repeatedly pulling on a linear crank (pull handle). The projectiles are three 1” diameter foam balls loaded into a hopper on top of the gun.
Buzz Saw Video

Step 1: User picks up the fully loaded blaster by the green-gripped handle.
Step 2: User firmly holds the green-gripped handle and pulls back repeatedly on the light blue charge handle until the desired amount of angular momentum is attained in the flywheel. This is an intuitive process for the majority of users. High angular speeds are indicated with a high pitch whining spooling noise from the flywheel and low angular speeds are indicated with a low pitch spooling noise.
Step 3: User steadies the gun and aims at target(s)
Step 4: User fires as many balls as desired, repeating steps 2 and 3 as necessary.
Step 5: User retrieves balls.
Step 6: User loads three balls into the hopper.

Summary
Each of the three guns has very different procedures for use. The Big Bad Bow has single-action priming and single firing. The Rapid Fire AS-20 has a significant amount of priming and preparation work for a rapid release of 20 darts or 20 single shots. The Buzz Saw requires spooling of a flywheel and offers 3 shots.

### User Study

A user study was conducted on the three Nerf blasters on Carnegie Mellon University's Pittsburgh Campus. Details on the nature of the experiment as well as observations and paraphrased quotes from the users can be viewed at the Nerf_gun_userstudy Wiki page.

General Feedback: People tended to initially think the gun was very appealing, but after using it were often disappointed in its performance.

User responses:
1) Sight was useful for lateral accuracy, not vertical
2) Large learning curve for an accurate shot
3) Not as exhilarating as other Nerf weapons
4) Ammo breaks easily
5) Operation is not immediately apparent, unclear on how to hold it
6) Short Limited range
7) Too large
8) Fulfills a classic "image"
Recognizable 9) Can not de-cock the gun without firing
10) Not immediately apparent how to cock it
11) Aesthetics and functionality do not align
User expected more function and interaction based on aesthetics; it's just a normal gun at its root

12.)Retrieving ammo is a hassle

##### Rapid Fire 20

General feedback: This weapon was praised as the most enjoyable to use, but there were frequent complaints people had about getting the gun ready to fire.

User responses:
2) Good for shooting crowds.
3) 20 rounds in 2.2 sec is impressive, unique, and exciting for most people
4) Not very comfortable
5) No pressure gauge, no user feedback
6) Pumping become difficult after 20. Pumping force is too great for small child
7) Tube behind grip is annoying
8) Hard to aim, there is no sight
9) Pump feels flimsy
10) More satisfying to fire than other Nerf guns
11) User gets feeling of quality with firing performance
12) Picking up all the fired ammo is tedious
13) "Single Fire" mode does not function properly, fires 4 to 5 darts instead of 1. Requires some finesse to fire one shot because enough air is let into the gun with one full squeeze of the trigger to fire off more than one shot.
14) Wasn't apparent at first that it needed to be pumped to fire
15) Not all of the darts fire; occasionally some get stuck or do not have enough pressure
16) Not aesthetically pleasing
17) Too violent for children according to some people
18) Unbalanced and cumbersome mass for a 7 year old
19) Pumping process is tedious

##### Buzz Saw

General feedback: This weapon is very different than a standard Nerf gun, and has a lot of annoying quirks because of it.

User Response:
1) Too much lag between trigger pull and ball firing
2) User loses time in steadying the gun after the winding phase (step 2) and before firing (step 4)
3) Sound is uncomfortable and prolonged for some users, but intimidating for the target
4) Sound does not accurately portray performance
5) User can vary projectile distance
6) Strength and experience limit gun performance
7) Runs out of ammunition quickly
8) “Buzz Saw” is a violent name
9) Cannot pump and fire at the same time. Balls get stuck in gun when one is pulling on light blue handle; The design includes a ball release switch to address this issue
10) Balls gets stuck in top of hopper
11) Gun does not shoot at an angle greater than about 45 degrees

Summary
Overall we observed a lot of interesting and entertaining features to the Buzz Saw and Rapid Fire for the user. Such features we expected of the Big Bad Bow but were not provided to the user, consequentially there was relative dissatisfaction with the Big Bad Bow. It looked like it performed, but failed to do so practically. For these reasons we decided to focus on improving the performance of the Big Bad Bow.

### Performance Studies

To further investigate the differences in performance between the guns we conducted experiments to test the distance and accuracy of the three guns.

#### Distance Study

Set Up: Guns were fired indoors from a horizontal orientation from 33.5" above the ground, and the position where the ammo first hit the ground was measured. The gun was held very steady when being shot so that no extra forward momentum was imparted on the projectile. Tests were performed indoors, so there was no wind or other disturbances. The Big Bad Bow and the Buzz Saw were each fired 9 times, then the distances were averaged to come up with the final values. The Rapid Fire was fired on multi shot twice, and it was given a spread.

Results:

Rapid Fire 20: 18' - 22' Spread
Buzz Saw: 30' 2"

Videos

#### Accuracy Study

Set Up: 3 people without previous Nerf gun experience were selected at random to participate in the study. Each person was instructed to stand 17' away from a target and fire with each separate Nerf gun at the target. Videos were taken in order to measure their accuracy, and are displayed below. Then one of our group members, after practicing with all three guns, did the same accuracy test and the results were recorded as well. After we are done processing the video from our group members, we will be able to figure out which guns have the steeper learning curves as well.

Results:

Rapid Fire: 55%
Buzz Saw: 100%

Example Videos

Videos with Results:

Conclusions: The distance study was very useful in giving the relative power of each weapon, since the distance each dart traveled is nearly proportional to initial velocity (neglecting air resistance). Because each trial was performed numerous times and averaged, the data is fairly robust. The accuracy tests also give a solid baseline about the general difficulty in aiming (basically that the Rapid Fire gun is the hardest to aim), but not much more than that. In order for the accuracy data to be more useful, many more tests would have to be performed by many more untrained subjects, and then averaged. By comparing the distances of projectiles from the two non rapid fire weapons, it is clear that the Big Bad Bow could be dramatically improved without having to worry too much about safety.

### Product Dissection

The Big Bad Bow proved to be mildly difficult to disassemble. Upon full disassembly it was apprent how refined and simplified the design was; there are only 27 seperable parts and 18 fasteners that make up the entire bow. Notable features in the disassembly were the use of only one kind of screw that were all inserted in the same direction. The gun also features almost all plastic construction.

### Bill of Materials for Big Bad Bow

Part No. Name Function Quantity Material Mass Manufacturing Process Picture
101 Bow Arm and Outer Cog Sub-Assembly Cosmetic 2 Plastic 56g Injection Molded
102 Small Inner Cog Cosmetic 2 Plastic 5g Injection Molded
201 Front Left Cover Priming Mechanism Housing 1 Plastic 117g Injection Molded
202 Front Right Cover Priming Mechanism Housing 1 Plastic 118g Injection Molded
203 Rear Right Cover Barrel/Action Housing 1 Plastic 123g Injection Molded
204 Front Left Cover Barrel/Action Housing 1 Plastic 119g Injection Molded
301 Nozzle Air Delivery/Dart Delivery 1 Plastic 17g Injection Molded
302 Nozzle Tip Air Restriction/FOD Protection 1 Plastic <1g Injection Molded
303 White Tip Holder Holds Nozzle Tip 1 Plastic 3g Injection Molded
304 White Air Restrictor + Spring Air Restriction 1 Plastic/Steel 3g Injection Molded
305 Rubber Plunger Air Delivery 1 Latex Rubber 5g Injection Molded
306 Barrel Lock Action Lock 2 Plastic 2g Injection Molded
307 Piston Spring Shaft Air Delivery 1 Plastic 26g Injection Molded
308 Trigger Lock Action Release 1 Plastic 2g Injection Molded
309 Trigger Action Release 1 Plastic 8g Injection Molded
310 Spring Stop Piston-Cylinder Spring Compression 1 Plastic 1g Injection Molded
311 Cylinder Air Delivery 1 Plastic 26g Injection Molded
401 Eyepiece Cosmetic 1 Plastic 4g Injection Molded
402 Eyepiece Lens Cosmetic/Aiming 1 Acrylic <1g Die Cut
501 Arrow Dart Ammo 3 Polyurethane 5g Molded/Glued
901 Screw Fastener 18 Steel 0.5g Forged/Machined
902 Spring for Barrel Lock Action Spring 2 Steel <.5g Rolled/Bent
903 Short Trigger Spring Release Mechanism 1 Steel <1g Rolled/Bent
904 Long Trigger Spring Release Mechanism 1 Steel <1g Rolled/Bent
905 Piston Spring Air Delivery 1 Steel 12g Rolled/Bent

## Preliminary Analysis

### Stakeholders

The major post-production stakeholders the product are the user, the parents of the user (if applicable), law-enforcement, and retailers. The parent has the most influence in the purchase of the product.

User (Child): The user of the product is often a child however the user is sometimes older than adolescent. For the child the major product needs are that the gun should:

• be aesthetically pleasing
• appear superior to other guns. This could include anything from “flamboyant” to “powerful”.
• operate according to the strength of a child
• weigh according to the strength of a child
• operate according to the bodily dimensions and ergonomics of a child
• require no instruction
• perform to impress other children. Performance could include remarkable functions, range, and rate of fire
• not break when thrown, dropped, or swung into things
• have easily retrievable or replaceable ammunition
• shoot with moderate accuracy
• shoot with moderate precision

Parent: The parents or guardians of the user will govern the use of the gun, if the user is a child. If the user is older than 18 years of age chances are that the user will usurp the parent’s needs as well as the user’s needs. Below are the needs of the parent. The gun should

• not require maintenance with normal use and lifespan
• cost less than $40 (might not apply for an older user) • appear non-threatening • have minimized commonalities in appearance and function with real guns or weapons for fear that it promotes violence. • not be injurious or dangerous to the user. For example the product should not have sharp edges, contain explosives or other flammables. • not be injurious or dangerous to others in proximity • excite and entertain the child • be non-toxic in body and ammunition • have easily locatable ammunition or inexpensive replaceable ammunition • be easy to clean up after use • be easy to store after use Law-Enforcement: The law enforcement authorities are also stakeholders seeing as they need to be aware that the gun is a toy and not a real gun. The gun must • have minimized commonalities in appearance and function with real guns or weapons. Gun can either be oddly shaped, have a flashy color, or have a bright orange cap on the barrel [Federal Regulation] • not be injurious or dangerous to others in proximity Retailers: Retailers are also involved as post-production stakeholders. Their needs are that the gun be • easy to stack and store on shelves • moderately sized • easy to sell • durable as to not require returns • satisfying to the user and parent so as to not require returns ### DFMA Disassembly of the Big Bad Bow revealed many considerations by the original manufacturer in their Design for Manufacturing and Assembly (DFMA). The first and most obvious choice in the design for manufacturing is the use of the same plastic material throughout almost every individual part of the Nerf blaster. External cosmetic components as well as the mechanical components in the firing mechanism are all made of the same type of plastic. This plastic comes in a variety of colors depending on the portion of the blaster it is used on. All of the parts made of this plastic showed obvious tell-tale separation lines and injection points from an injection molding process. Injection molding appears to be the most efficient and cost effective method of manufacturing the parts for Nerf guns as Hasbro Inc. shows over$1 Billion in revenue per year indicating that the sales in Nerf brand toys to be in the millions of units. Injection molding should allow a high rate of manufacturing after initial tooling and assembly line setup at a low price and short turnover per unit. The ability to produce injection molded parts also allows the unit to be built in a variety of colors and in very complex shapes, adding to the overall aesthetic appeal of the product and shortening the production time by eliminating paint or coatings. A final notable design-for-manufacturing point is that the darts are made of a molded polyurethane foam, another choice that would allow the part to be made in bulk and at a very high rate, at low cost.

The most apparent optimization in the DFMA considerations of the Nerf Big Bad Bow are in its design for assembly. The entire unit can be assembled from one side with one type of hardware and only one tool. The entire product is assembled starting with the right cover down, then all internal components are placed; finally, the left cover placed to complete the assembly. The assembly can be broken down into two sub-assemblies; there is the barrel assembly and the trigger assembly. The barrel assembly houses the piston mechanism and the firing barrel as well as the gun tip. The trigger assembly includes parts like the trigger itself, the release mechanism, the trigger springs and trigger lock. Both of these sub assemblies fit together in between the left and right rear covers. The assembly can be seen in the image here:

The final left and right front covers are then used to enclose the assemblies. The covers actually act as assembly housings and are necessary parts of the firing mechanism in addition to being a cosmetic cover. This multifunctional part allows the manufacturer to eliminate extra parts and simplify manufacturing and assembly.

All fastening is done with the same type of self-tapping screws; all of the screws go through holes in the right panels and into bosses on the left panels. This allows the assembler to approach the product from only one side while assembling. Glue was also used to fasten the left and right rear covers around the barrel assembly. Plastic studs and bosses allowed the left and right rear covers to self align before they were glued.

After the front panels are assembled, all the cosmetic elements of the product such as the bow and the bow string cogs snap into place with various plastic bosses and holes in the parts; a portion of this assembly is actually left to the end-user to allow the Nerf Big Bad Bow to fit in its packaging.

Overall, the choices apparent in the DFMA of the Nerf Big Bad Bow show that Hasbro has taken extensive measures to optimize the speed of production, minimize parts and materials, simplify design, and utilize multifunctional parts. This should allow the Big Bad Bow to be produced extremely cheaply and sold comparatively cheap.

### FMEA

Determining where the Nerf guns could fall short of intended function is of utmost importance for redesign. The Big Bad Bow functions according to the company’s intentions. The company must demonstrate a balance between performance and cost. In order minimize manufacturing costs, shortcuts are taken in the product design.

These shortcuts bring up possible failure modes for each part. We examine each part for modes of failure that effect the consumer and devise possible remedies for those failures by the company. This is our Failure Mode and Effects Analysis (FMEA). Due to the size of the document it can be found by following this link Big Bad Bow FMEA

The algorithm for the analysis is adapted from a Ford Motor Company methodology cited from “Engineering Design” by Dieter and Schmidt. The part numbers correspond with the parts listed in the Nerf_gun#Bill_of_Materials_for_Big_Bad_Bow. The S refers to a rating of Severity of Failure. The O refers to a Rating for Occurrence of Failure. The D refers to a Rating for Detection of Failure. RPN refers to the Risk Priority Number which multiplication of S, O, and D. Refer to the Dieter & Schmidt textbook for descriptions for each rating.

After inspecting possible failure modes for each part we determined we would focus on the following issues.

• Remedying ammunition construction failures (part number 501)
• Improving upon firing mechanisms (part numbers 301 to 311)
• Making aesthetic parts to be functional. (part numbers 101, 102, 201-204, 401, and 402)

### DFE

Creating and selling a product has a significant environmental impact from increased production activity in it's sector and associated sector. Because the Nerf gun does not require energy input for uses or additional accessory products for use, the environmental impact analysis focus on its production and end-of-life impact. In order to measure the environmental impact of production of a Nerf gun, we used the Carnegie Mellon University's Green Design Institute EIO-LCA tool to quantify the increase in greenhouse gas production, conventional air pollution, and toxic releases caused by \$1 million in sales in the doll, toy, and game manufacturing sector. Based on the results of the EIO-LCA, designing the Nerf gun to use low energy, low toxic byproduct manufacturing processes would significantly reduce the environmental impact of the Nerf gun production process. At the end of a Nerf gun's life, the product is generally disposed of by the consumer. Most likely, the consumer will throw the Nerf gun in the garbage, and it will be taken to a landfill. Since a Nerf gun is made mostly from plastic, it will not easily decay, and as it decays, will potential release toxins to the environment. To reduce the end-of-life environmental impact of the Nerf gun, it should be designed to be recyclable. We are in the process of determine what type of plastic is used in the Nerf gun. If this plastic is not recyclable, the gun could be re-designed to use a recyclable plastic. The Nerf gun should also be designed to encourage the consumer to recycle the product. If the Nerf gun and its packaging included messages informing the consumer that the consumer that it is recyclable and encourage them to recycle, it would increase recycling of the product, thereby reducing its end-of-life environmental impact.

We believe that the category for 'doll, toy, and game manufacturing' sector is where a Nerf weapons belongs in this test. This category includes other items such as toy rifles, craft and hobby kits and sets, model railroads, toy furniture and household-type equipment, doll carriages, and children's scooters. Most of these items are made with very similar materials as Nerf weapons (lots of plastic with different types of hardware). In addition, the presence of a toy rifle at the beginning of this list implies very strongly that Nerf weapons belong in this category as well.

Greenhouse Gas Emissions

Greenhouse gas emissions are a particularly important consideration in design for environment due to the current concern with global warming and the likelihood of carbon taxes. The EIO-LCA analysis shows that energy generation accounts for 45% of the greenhouse gas emissions associated with doll, toy, and game manufacturing. Using a less energy-intense manufacturing process for the Nerf gun would reduce greenhouse gas emissions associated with its production.

The chart below summarizes the greenhouse gas emissions of the top 10 sectors related with doll, toy, and game manufacturing sector.

The chart below depicts the contribution of each top 10 sectors related with doll, toy, and game manufacturing sector to the total greenhouse gas emissions associated with doll, toy, and game manufacturing.

Conventional Air Pollutants

Conventional air pollution are an important consideration due to its impact on air quality and acid rain. The EIO-LCA analysis shows that energy generation accounts for 80% of the conventional air pollutant emissions associated with doll, toy, and game manufacturing. Using a less energy-intense manufacturing process for the Nerf gun would significantly reduce conventional air pollutant emissions associated with its production.

The chart below summarizes the conventional air pollutant emissions of the top 10 sectors related with doll, toy, and game manufacturing sector.

The chart below depicts the contribution of each top 10 sectors related with doll, toy, and game manufacturing sector to the total conventional air pollutant emissions associated with doll, toy, and game manufacturing.

Toxic Releases

Toxic releases are an important consideration due to their hazard to human, animal, and environmental health. The EIO-LCA analysis shows that plastic and resin manufacturing and doll, toy, and game manufacturing each account for 26% of the toxic releases associated with doll, toy, and game manufacturing. Using a plastic that produces minimal toxic byproducts in the Nerf gun, and using manufacturing processes with low levels of toxic byproducts would significantly reduce the amount of toxic releases associated with its production.

The chart below summarizes the toxic releases of the top 10 sectors related with doll, toy, and game manufacturing sector.

The chart below depicts the contribution of each top 10 sectors related with doll, toy, and game manufacturing sector to the total toxic releases associated with doll, toy, and game manufacturing.

### Mechanical Analysis

#### Mechanical Operations

The entire firing mechanism is fitted into the barrel/action housing. The piston shaft is held in place by the action release. When the barrel is pulled back by the user, the action release pulls the piston shaft with it compressing the spring in the cylinder which is held in place by the main gun body. The trigger, when pulled, lifts the action release and lets go of the piston shaft which, pulled forward the compressed spring, rams the air in the cylinder forward through the nozzle and into the dart which is then flung from the nozzle and hopefully hits the target.

#### Mathematical Analysis

Many users complained that they were disappointed with the firing power of the Big Bad Bow and felt that it should be able to fire darts further. In order to determine the modifications that should be made to the firing system in order to increase the distance fired, the maximum firing distance of the Big Bad Bow must be determined. To determine the maximum firing distance, we need to determine the maximum force applied by the gun to the dart in an ideal case. This analysis can be used to determine how altering parameters such as spring stiffness and barrel pressure will impact the firing distance.

Under ideal conditions, the following assumptions will hold:

1. No energy is lost from the system to to heat, friction, sound, etc.
2. The spring used in the firing mechanism is a linear spring and is not stretched outside of it's elastic region during firing (Hooke's Law can thus be applied)
3. The only force acting on the dart after is is fired is gravity
4. The air in the firing chamber is an ideal gas (Ideal Gas Law can thus be applied)
5. The temperature of the air in the firing chamber is constant
6. The air pressure in the firing chamber is uniform
7. The firing chamber is sealed, there is no pressure loss in the chamber due to air leakage

The spring used in the firing system system provides force to push to compress the air in the cylinder, increasing the pressure. The air pressure then applies force to the dart. Because there is no energy loss in the system through transfer of energy from the spring to the air or from the air to the dart, the energy can be treated as being transfer from the spring directly to the dart for the purpose of analysis.

Energy applied by the spring is given by the equation:

$\ U_s = \tfrac{1}{2}kx^2\$

where Us is the spring energy, k is the spring constant, and x is the distance that the spring is compressed.

Kinetic energy of the dart upon firing is given by the equation:

$\ T = \tfrac{1}{2}mv^2\$

where T is the kinetic energy of the dart, m is the mass of the dart, and v is the initial velocity of the dart.

Applying Conservation of Energy: $\ T = U_s \$

$\tfrac{1}{2}mv^2 = \tfrac{1}{2}kx^2 \$

$\ v = \sqrt{\frac{kx^2}{m}}$

From measurements taken of the Big Bad Bow: $\ x = 0.087m \$ $\ m = 0.005kg \$

The spring skiffness is unknown and must be determined experimentally. To determine the spring stiffness, a 5lb rock was placed on the spring and it's compression was measured. The experiment yielded the following data:

• rock mass: $\ m_r = 2.26kg \$
• initial spring length: $x_o = 0.121m \$
• compressed spring length: $x_f = 0.082m \$

The spring stiffness can then be calculated using Hooke's Law:

$\ F = kx \$ $\ k = F/x \$

where F is the spring force and x is the amount by which the spring is compressed.

The spring force can be determined by applying Newton's Law:

$\ F = m_r g \$


where g is the gravitational constant, g = 9.8m / s2.

Using the experimental data in the formulas above, the spring stiffness is found to be:

Failed to parse (unknown function\k): \k = 570\frac{N}{m} \

Using the Big Bad Bow's spring stiffness, dart mass, and the distance the spring is compressed, the initial velocity of the dart is found to be:

$\ v = 29.4\frac{m}{s} \$

Using kinematic equations for projectile motion, we can determine the maximum distance the dart could be fired under conditions identical to those in performance testing of the Big Bad Bow.

To calculate the maximum distance fired by the Big Bad Bow in conditions identical to our tests we use the following experimentally determined information:

Vertical motion

• initial height: $\ y_0 = 0.85m \$
• final height: $\ y = 0m \$
• initial velocity: $\ v_y0 = 0\frac{m}{s} \$
• final velocity: $\ v_y = unknown \$
• acceleration: $\ a_y = g = 9.8\frac{m}{s^2} \$

Horizontal motion

• initial position: $\ x_0 = 0m \$
• final position: $\ x = unknown \$
• initial velocity: $\ v_x0 = 29.4\frac{m}{s} \$
• final velocity: $\ v_x = 0\frac{m}{s} \$
• acceleration: $\ a_x = 0\frac{m}{s^2} \$

time: $\ t = unknown \$ is the same for both horizontal and vertical motion.

The time can be calculated from the vertical motion:

$\ y-y_0 = v_0y t + \tfrac{1}{2}a_y t^2 \$

$\ t = \sqrt{\frac{2y_0}{a_y}} \$

$\ t = 0.59s \$

The distance travel can then be calculated from the horizontal motion:

$\ x - x_0 = \tfrac{1}{2}v_0 t + \tfrac{1}{2}v t \$

$\ x = \tfrac{1}{2}v_0 t \$

$\ x = 8.63m \$

The projectile motion calculations show that the maximum distance fired for the dart is larger (by about 3.5m) than the actually distance fired by the Big Bad Bow. Energy loss due to friction in the transfer of energy from spring to air to dart, to losses in air pressure due to air leakage, and to effects of drag on the dart.

The calculations below are used to calculate air pressure in the cylinder. This information is used to create graphs showing the relationship between spring constant, air pressure, and force applied to the dart. These graphs can be used later to determine impacts of modification of spring constant and cylinder air pressure on the firing distance of which the gun is theoretically capable.

First, the initial volume of the cylinder needs to be determined. The cylinder is as long as the spring with a diameter of 2.7 cm.

$V = (.027m)^2 \cdot \pi /4 \cdot .121m = 6.928 \cdot 10^-5 m^3$

Using the Ideal Gas Law:

$P \cdot V = n \cdot R \cdot T$

Since the initial volume and pressure and temperature are known, calculating the moles of gas in the cylinder is simple. Then, we must find the volume as a function of the position of the plunger.

$V = V_0 - A \cdot l$

Where V_0 is the initial volume, A is the area of the cylinder and l is the position of the plunger from the pulled back position. So when l = 0, the plunger is all the way back.

Now the Pressure as a function of Volume can be found.

$P = \frac{n \cdot R \cdot T}{V}$

We can plot internal pressure of the cylinder as a function of the plunger position, and then so can the force on the cross section of the nozzle as a function of internal pressure and finally force as a function of the position of the plunger.

Mechanical Operations Concluding Notes

The actual propelling force on the dart will be higher and will involve the force of the volume of air in the cylinder being pushed into the space inside the dart, but the point is clear; already there is a tremendous decrease in force between the spring and the dart. This explains the relatively short distance the dart travels. This is done, most likely, so that the Big Bad Bow is safe for little kids to play with.

To make this a toy that would be fun for people in our age group, the force directly applied to the dart would have to be much bigger, or the ammunition would have to have a longer time to accelerate. This means that the mechanism for launching the darts needs to be modified or changed entirely. Modifications would include any modification that would substantially increase the pressure in the cylinder prior to launch, such as using a compressed air system (CO2 catridge, Green gas, etc) to launched the darts. A change to the entire launch mechanism would involve either using some sort of rail system to give the dart more time to accelerate or applying the launching force directly from the energy storage element (like a real bow or crossbow) or some other system not yet mentioned.