# Ballistics Demonstrator The science of ballistics involves the motion of projectiles. Exterior ballistics concerns the projectile after it leaves the device that it is shot from. The term "ballistics" comes from the name of one kind of catapult, the ballista.

## Achievement

1. Identify the Problem
• Write a synopsis of the problem in your engineer's notebook such that you could finish this challenge without further reference to this web site.
• Be sure and include all criteria and constraints.
• Ask questions as needed to clarify these directions.
2. Generate Concepts
• Sketch ideas in your notebook.
• Sketch an improved version of your best idea on a separate paper.
• Combine your sketches with those of the other members of your team and use a decision matrix to choose a design to develop further.
3. Develop a Solution
• Create a set of working drawings in your notebook in sufficient detail that a third party could recreate your chosen design based solely on what you have in your notebook.
4. Construct and Test a Prototype
• Construct a device from scrap material that will throw a marshmallow at least 15 feet.
• Shoot six times at an elevation angle of 45°, measuring the range (distance) of each shot. Use this information to estimate the intitial velocity.
• Based on this initial velocity, predict the range at 30° and 60°.
• Shoot six times each at 30° and 60° to check your prediction.
• Demonstrate the performance of your device by shooting three times at a one foot diameter target from at least 15 feet away, without any practice shots. Your score will be based on the total distance from the target for your three shots.
5. Evaluate the Solution
• Calculate the median, mean, mode, and standard deviation of each data set.
• In your engineer's notebook, compare your results to others in the class and explain why your device did better or worse than others.
6. Present the Soltution
• Document each step in the design process in your own engineer's notebook
• Do the problems on GUESS paper.

## Constraints

• Teams of 2 or 3.
• Energy to throw the marshmallow must come from energy stored in rubber bands, twisted cordage, or weights.
• No more than 24" in any dimension.
• Must shoot at different angles (at least 30°, 45°, and 60°) at the same initial velocity
• The mean range at 45° must be at least 15 feet.
• Cannot practice at the target.

## Standards

### Engineer's Notebook—10 points

Evaluated using the usual rubric. Make sure you:

• Have numbered each page sequentially (outside top). If not, do it!
• Fill each page on a continual bases without large blank areas. "A page a day" is not the system here.
• Glue or staple in all inserted items.
• Sign the bottom of each page when full. Date each signature (the date of the entry is not sufficient).
• Date each entry.
• Obtain at least one witness signature. This signature should be dated (the date of the entry is not sufficient).
• Sketch your device. Do your calculations in the notebook. Annotate the sketches and calculation so that you will be able to figure them out in a year or two.
• Document every step in the design process. Clearly explain how your design was supposed to work.
• Document the results of each data run. When you encounter problems, as you surely will, write down your ideas to fix them.
• Write down everything that you do as you do it so that the information in your notebook is proportional to the amount of time spent on this project.

### The Device—10 points

The device will be judged based on the quality of construction and the design elements that let it shoot a marshmallow a consistent and repeatable distance.

### Performance During Calibration—20 points.

5 points each for:

• Computing the intitial velocity at 45°
• Predicting the range at 30° and 60°
• Shooting at 30° and 60°
• Calculating the median, mean, mode, and standard deviation of each data set.

### Performance at the Target—20 points.

Your score is 20 points minus your total distance away from the target in feet.

### Completition of the Problems—20 points

Make sure and:

• List the given and unknown quatities as symbol=number unit.
• Show each equation.
• Substitute numbers and units for symbols in the equations.
• Solve each equation for the desired quantities.
• Carry units throughout.
• Use g = -32.2 ft/sec2

### Ballistics Formulae (see derivation below)

Range X = Vi2sin2θ/(-g)

Altitude Ymax = Vi2sin2θ/(-2g)

Time of Flight tof = 2Visinθ/(-g)

## The Problems

1. (3 points) A skydiver jumps from 10,000 ft. How fast is she going after:
a. 1 second?
b. 2 seconds?
c. 8 seconds?
d. What is her altitude after 8 seconds?

2. (3 Points) A marshmallow is dropped from the roof of the school, 65 feet from the ground. How fast is it going and what is its altitude after:
a. 1 second?
b. 2 seconds?
c. 3 seconds?
d. What is her altitude after 8 seconds?

3. (4 Points) A baseball is thrown straight up at 64.4 ft/sec.
a. How long does it take to get to its highest point?
b. How high does it go?
c. How long does it take to get back to your glove?
d. How fast is it going when you catch it?

4. (3 Points) A projectile is thrown at 100 ft/sec at an elevation angle of 45°.
a. How far does it go (range)?
b. What is the maximum altitude?
c. What is the time of flight?

5. (3 Points) A projectile is thrown at 100 ft/sec at an elevation angle of 30°.
a. How far does it go (range)?
b. What is the maximum altitude?
c. What is the time of flight?

6. (4 Points) A projectile is thrown at 100 ft/sec at an elevation angle of 60°.
a. How far does it go (range)?
b. How does this range compare to the range at 30�?
c. What is the maximum altitude?
d. What is the time of flight?
Bonus: at which angle did the projectile go farthest? Why don't hit baseballs fly the farthest at this angle?

# External Ballistics Formulae

## General Equations Under Constant Acceleration

Acceleration (a) = constant = g = -32.2 ft/sec2 or -9.8 m/sec2

Velocity is then equal to the intitial velocity (Vi or sometimes V0) plus the acceleration times time.

Velocity (V) = Vi + gt

Position is the initial position (d0), plus the initial velocity times time, plus one half acceleration times time squared.

Position (d) = d0 + Vit + ½gt2

## External Ballistics Input Variables

The range, altitude, etc. of a projectile depend only on the initial velocity (Vi or sometimes written V0) and the elevation angle (Greek Theta, θ)

The initial velocity vector can be broken into its X and Y components where:

Velocity in the Y (vertical) direction = Visinθ

Velocity in the X (horizontal) direction = Vicosθ

## Initial Conditions

In the Y (vertical) direction the equations become:

Velocity (VY) = Visinθ + gt

Altitude (Y) = Y0 + Visinθ t + ½gt2

In the X (horizontal) direction there is no acceleration. Assume the projectile starts at the origin. It will move at a constant speed in X and then:

Velocity (VX) = Vicosθ

Position (X) = Vicosθ t

## Dropping a Ball (or other projectile)

The initial velocity is 0 and the ball accelerates downward, gaining 32.2 feet per second of downward velocity each second.

Velocity (VY) = gt

The altitude will be the original altitude (Y0) plus one half g times time squared. But since g in negative, the altiude will go down.

Altitude (Y) = Y0 + ½gt2

Note that g is a negative number and that the time is squared in the altitude formula.

## Throwing a Ball Straight Up

In this case, the initial altitude (Y0) is 0. The ball will go up, slowing down until it stops at the top. The ball will then accelerated downward and return to your hand with same speed that it left, but in the opposite direction. The initial velocity is Visinθ, but since the sine of 90° is 1, the initial velocity is just Vi

How long does it take to get to the top? At the highest altitude V = 0

V = 0 = Vi + gt

-gt = Vi

ttop = Vi/(-g)

What is the maximum altitude (Ymax)? Put the time to the top in the altitude equation.

Altitude (Y) = Y0 + Vi t + ½gt2

Ymax = Y0 + Vi t + ½gt2

Set Y0 to 0 and replace t with Vi/(-g)

Ymax = Vi Vi/(-g) + ½g[Vi/(-g)]2

Ymax = Vi2/(-g) + ½[g/(-g)2]Vi2

Ymax = -Vi2/g + ½Vi2/g

Ymax = ½Vi2/(-g) or Vi2/(-2g)

In the general case, ball not thrown straight up:

ttop = Visinθ/(-g)

Ymax = Vi2sin2θ/(-2g)

## Time of Flight

When the ball gets back to your hand the altitude is once again 0.

Altitude (Y) = Y0 + Vi t + ½gt2

0 = 0 + Vi t + ½gt2

Factor out t on the right

0 = t [Vi + ½gt]

This equation has two solutions, one then t = 0 and then when:

0 = Vi + ½gt

-½gt = Vi

And the time of flight tof = 2Vi/(-g), which is twice the time to the top

In the general case, ball not thrown straight up: tof = 2Visinθ/(-g)

## Launching a Projectile at an Angle. What is the range?

There is no acceleration in X. If you set X0 to 0, the equation becomes:

X = Vicosθ t

Substituting the time of flight for t

X = Vicosθ2Visinθ/(-g)

Rearranging

X = Vi22cosθsinθ/(-g)

But 2cosθsinθ is equal to sin2θ

Range (X) = Vi2sin2θ/(-g)

## In Summary

Range X = Vi2sin2θ/(-g)

Altitude Ymax = Vi2sin2θ/(-2g)

Time of Flight tof = 2Visinθ/(-g)

## Here is a Table of Common Sines and Cosines

Anglesincossin2cos2sin2θ
30°0.50.870.250.750.87
45°0.7070.7070.50.51
60°0.870.50.750.250.87
90°10100 