Real rocket data

For typical rocket with A engine:

Apogee height 57 m
Top speed 34 m/s
Thrust time 0.5 sec
Peak acc 22 g's
Average acc 7.1 g's
Time to coast after burn out until apogee 3.7 sec
Descent rate 5 m/s
Total time of flight 14 sec

You can get approximate values for b and c engines by doubling and tripling these numbers.

hw for Tuesday

Review all the recent material in preparation for the take-home test:

1.  One dimension motion
2.  Projectile motion
3.  River problems
4.  Newton's laws and applications of F = ma

You are permitted to use your notes on the take-home test.  I'll give it to you on Tuesday and it will be due Thursday.

Rocket stuff, if you're looking ahead.

Rocket calculations, if you're working ahead.

 Rocket Lab

In this lab, you will determine the following information about your rocket flight:

Maximum speed

Maximum height

Time of flight

1.     Before the flight, record the mass (in kg) of your rocket.  The scale will give it to you in grams – convert to kg.  The rocket should have everything inside of it:  wadding, engine, parachute.

2.     Record the type of engine being used.

3.     The average thrust (Force, F) is the first number (probably 8 or 6) in the above engine type.  This number is in newtons.  Record here:

4.     Also record the estimated height (in m), using an altimeter or trigonometry. 
(WE WILL DO THIS ON LAUNCH DAY.)

5.     Determine the post-flight mass.  Subtract the following numbers from your pre-launch mass.  If the engine is different, ask Sean

A8-3 engine (3.12 g = 0.00312 kg).  B6-4 engine (6.24 g = 0.00624 kg)

6.     Now find the average mass (between pre and post launch masses).

7.     Calculate the average acceleration of the rocket during its thrust period.

8.     The time of thrust is known - it is set by the amount of propellant in the engine.  A8-3 (0.5 sec). B6-4 (0.8 sec).  See Sean if engine is different.

9.   Find the "burnout speed" of the rocket, using an equation of motion.

10.   Determine the height to which the rocket has climbed by this point.

11.   Now, consider the burnout speed as the (new) initial speed and find the height to which the rocket will continue to climb until it reaches apogee.

12.   Find the total height (theoretically) achieved.

13.  Compare this to the estimated height from launch day.  See 4 above.  

14.  What is the discrepancy between the estimated height (5 above) and the calculated height (13 above).  Why are the numbers different?  Discuss.

15.  Draw a labeled picture that represents the flight of your rocket.

Newton HW

1.  Consider a 0.05 kg model rocket.

a.  If it has an engine attached that provides 6-N of thrust, what is the rocket's acceleration?
b.  If the thrust time is 0.5 seconds, what is the final "burn out" velocity?
c.  What will happen after "burn out"?

2.  These questions are about mass and weight (the force due to gravity)?

a.  What exactly is the difference between mass and weight?
b.  What is the weight of a 65 kg woman?
c.  How do weight and mass change/compare on the Moon?

3.  Imagine standing on a scale in an elevator.

a.  How would the scale reading change if you were moving UP with a constant velocity?
b.  How would the scale reading change if you were moving DOWN with a constant velocity?
c.  How would the scale reading change if you were accelerating up with a constant acceleration?
d.  How would the scale reading change if you were accelerating down with a constant acceleration?
e.  You have a mass of 80 kg.  If were in a moving elevator, accelerating UP at 1 m/s ^2, what would the scale read?
f.  What would the scale read if you were in a freely-falling elevator?

4.  Give an example of Newton's 3rd law in action.

5.  Why do all objects fall toward the Earth with the same acceleration (in the absence of air resistance)?  Answer this question in light of Newton's 2nd law.

Newton's Laws redux.

1. Newton's First Law (Inertia) 

An object will keep doing what it is doing, unless there is a reason for it to do otherwise.

That means, it will stay at rest OR it will keep moving (at a constant velocity) unless acted on by an unbalanced force.

2. Newton's Second Law

An unbalanced force (F) causes an object to accelerate (a).

That means, if you apply a force to an object (and the force is unbalanced - greater than any resisting forces), the object will accelerate.

Symbolically:

F = m a

The Force (F) on a mass (m) produces acceleration (a), predicted by the above equation. In detail:

Greater F means greater a
If the Force is kept constant, but the mass is increased, the acceleration will be smaller:

a = F/m

That's an inverse relationship.


There is a new unit for Force - since Force = mass times acceleration, the units are:

kg m/s^2

We give this a new name, the newton (N). It's about 0.22 lb.




3. Newton's 3rd Law

To every action there is opposed an equal reaction. Forces always exist in pairs. Examples:

You move forward by pushing backward on the Earth - the Earth pushes YOU forward.

A rocket engine pushes hot gases out of one end - the gases push the rocket forward.

If you fire a rifle or pistol, the firearm "kicks" back on you.

Since the two objects experience the same force:

m A = M a

That's a little tricky to convey in letters but, the larger object (M) will experience the smaller acceleration (a) and the smaller object (m) will have a larger acceleration (A).

Newton's Laws 

Newton and his laws of motion.

Newton, Philosophiae naturalis principia mathematica (1687) Translated by Andrew Motte (1729)

Lex. I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.


Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.


Lex. II. Mutationem motus proportionalem esse vi motrici impressae, & fieri secundum lineam rectam qua vis illa imprimitur.


The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.


If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.


Lex. III. Actioni contrariam semper & aequalem esse reactionent: sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi.


To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.


Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I  may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other.

Derivations

Try these. Sorry for the late posts.

hw

1.  Revisit the river problem.  You are headed west across a river that flows south.  The speed of your boat is 15 m/s.  The river current is 6 m/s.  If the river is 120 m wide (east-west):

a.  how long does it take to cross?
b.  where do you land?

2.  Projectiles at angles.  If you were to throw a baseball at 30 m/s at an initial angle of 55 degrees:

a.  how long (time) would the ball remain in air until caught (at the same vertical height from which it was released)?
b.  how far horizontally would that ball travel during this time?
c.  (Tricky part.)  How high did the ball rise?  Hint:  It reaches apogee at the "half-way" point, if the problem is symmetrical (as it is in this case).

*3.  For fun.  Try to derive the range equation:

R = [ vi^2 / g ] sin (2 theta)

R is the range (also known as the horizontal displacement, dH).

It is useful to know the double angle relation:

sin (2 theta) = 2 sin(theta) cos(theta)

Hint:  Set up an equation for dH and plug in something for time.

This is not a trivial problem, so don't get frustrated if you can't get it.

HW

Using the cards from the other group, create a scale map of the treasure hunt. You'll need to share the data between group members. I suggest graph paper or a drawing program.

Use trigonometry to calculate the total displacement from start to finish, including angle. SOH CAH TOA will be useful, along with the Pythagorean theorem.

Also, use google maps or a similar site to create a map from where you live to school. Print it out, making sure that there is a scale somewhere on the map. Also make sure the North is indicated.

HW

1.  Consider the informal lab from today.  If you launched a ball from a height of 12-m, and it traveled 25-m, how fast was it traveling?

2.  Thinking in general terms about projectiles:

a.  If you were to double the initial speed of the projectile (launched horizontally), what effect would it have on the horizontal displacement (where it lands)?

b.  If you were to double the initial vertical height of the projectile (but keep the speed constant), what effect would this have on the time in air?

c.  Go back to 1-D motion for a bit.  Consider dropping an object from a certain height on the Earth vs. the same object at the same height on the Moon.  How do the drop times compare?  Calculate, if possible, the extent to which they are different (ie., one time is how many times the other time, etc.).

3.  Finally, have a look at the work of Isaac Newton.  List some of his noteworthy accomplishments.

One more HW question, if you see this in time

Consider a ball that was launched horizontally from a rooftop.  The initial height above the ground was 15-m.  It landed 20-m from the base of the building.  What was the initial speed of the ball?

One other thing to think about.  How would we go about dealing with launches at angles (which, of course, make up the bulk of projectile problems)?

Cool.

http://www.latimes.com/news/science/sciencenow/la-sci-sn-ibm-world-smallest-movie-atoms-20130501,0,5172409.story

http://www.forbiddenknowledgetv.com/videos/science/have-you-ever-seen-an-atom.html

Projectiles!

Consider the material from today's class, particularly the demo wherein I launched and dropped balls simultaneously. Answers these questions based on that scenario.

1. Why did they land at the same time?

2. Can you imagine a situation in which they would have not landed simultaneously? Discuss.

3. If the projected ball had been launched from a height of 6-m, with an initial speed (horizontally) of 8 m/s:

A. How long would it take to hit the ground?

B. How far horizontally would it go?

4. We will have an informal lab on Thursday. You will have a ramp that allows a ball to be launched horizontally. Propose a way to experimentally determine its initial speed. You will not be able to use a photogate.

Recent notes FYI

Velocity and Acceleration

Motion!

THE EQUATIONS OF MOTION!

First, let's look at some definitions.

Average velocity

v = d / t

That is, displacement divided by time.

Another way to compute average velocity:

v = (vi + vf) / 2

where vi is the initial velocity, and vf is the final (or current) velocity.

Average velocity should be distinguished from instantaneous velocity (what you get from a speedometer):

v(inst) = d / t, where t is a very, very, very tiny time interval. There's more to be said about this sort of thing, and that's where calculus begins.

Now this idea (velocity) is pretty useful if you care about the velocity at a specific time OR the average velocity for a trip.

Some velocities to ponder....

Approximately....

Keep in mind that 1 m/s is approximately 2 miles/hour.

Your walking speed to class - 1-2 m/s
Running speed - 5-7 m/s
Car speed (highway) - 30 m/s
Professional baseball throwing speed - 45 m/s
Terminal velocity of skydiver - 55 m/s
Speed skiing - 60 m/s
Speed of sound (in air) - 340 m/s
Bullet speed (typical) - 900 m/s
Satellite speed (in orbit) - 6200 m/s
Escape velocity of Earth - 11,200 m/s
(That's around 7 miles per second, or 11.2 km/s)

Speed of light (in a vacuum) -

c = 299,792,458 m/s

This number is a physical constant, believed to be true everywhere in the universe. The letter c is used to represent the value being of constant celerity (speed).


On the other hand, if you care about the details of velocity, if and when it changes, then we need to introduce a new concept: acceleration.

>

Acceleration, a

a = (change in velocity) / time

a = (vf - vi) / t

The units here are m/s^2, or m/s/s.

Acceleration is a measure of how quickly you change your speed - that is, it's a measure of 'change in speed' per time. Imagine if you got in a car and floored it, then could watch your speedometer. Imagine now that you get up to 10 miles/hr (MPH) after 1 second, 20 MPH by the 2nd second, 30 MPH by the 3rd second, and so on. This would give you an acceleration of:

10 MPH per second. That's not a super convenient unit, but you get the idea (I hope!).

>

Today we will chat about the equations of motion. There are 5 useful expressions that relate the variables in questions:

vi - initial velocity. Note that the i is a subscript.
vf - velocity after some period of time
a - acceleration
t - time
d - displacement

Now these equations are a little tricky to come up with - we can derive them in class, if you like. (Remember, never drink and derive. But anyway....)

We start with 3 definitions, two of which are for average velocity:

v (avg) = d / t

v (avg) = (vi + vf) / 2

and the definition of acceleration:

a = (change in v) / t or

a = (vf - vi) / t

Through the miracle of algebra, these can be manipulated (details shown, if you like) to come up with:

vf = vi + at

d = 0.5 (vi + vf) t

d = vi t + 0.5 at^2

vf^2 = vi^2 + 2ad

d = vf t - 0.5 at^2

Note that in each of the 5 equations, one main variable is absent. Each equation is true - indeed, they are the logical result of our definitions - however, each is not always helpful or relevant. The expression you use will depend on the situation.


In general, I find these most useful:

vf = vi + at

d = 0.5 (vi + vf) t

d = vi t + 0.5 at^2

By the way, note that the 2nd equation above is the SAME THING as saying distance equals average velocity [0.5 (vi + vf)] multiplied by time.


Let's look at a sample problem:

Consider a car, starting from rest. It accelerates uniformly (meaning that the acceleration remains a constant value) at 1.5 m/s^2 for 7 seconds. Find the following:

- the speed of the car after 7 seconds
- how far the car has traveled after 7 seconds

Then, the driver applies the brakes and brings the car to a halt in 3 seconds. Find:

- the acceleration of the car in this time
- the distance that the car travels during this time


Got it? Hurray!