More practice problems (and answers to last questions)

Answers to recent Newton problems:

1.  See notes.

2.  40 / 0.5 = 80 m/s/s

3.  lower acceleration

4.  See notes.  Tablecloth pull, etc.

5.  firearm recoil, etc.

6.  Principia Mathematica, 1687.

7.  They helped explain retrograde motion - the apparent backwards motion of planets.  Really, planets are orbiting the Sun and there are times that some bodies are "behind" -- it's like when you pass someone on the highway and they "appear" to be moving backwards at that time.

8.  mass - the amount of stuff (in kg); weight - the gravitational pull on this stuff (in newtons).  The weight depends on where you are (in terms of how the gravitational acceleration changes).  For example, your weight on the Moon is 1/6 that of Earth.

9.  newton; pound

10.  W = m g.  Weight is depending on the local value of g.


11.  Letting g = 10 m/s/s -->  40 m/s, 80 m

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New questions (and answers):

1.  Explain the meaning of "inverse square law".

2.  Discuss each of Kepler's 3 laws.

3.  At what point in its orbit is the Earth closest to the Sun?

4.  At what point in its orbit is the Earth moving fastest?

5.  What causes seasons?

6.  What is a semi-major axis of orbit (a)?

7.  What is an Astronomical Unit (AU)?

8.  Consider Jupiter.  It's orbit is 5 AU in size (roughly).  How long should it take Jupiter to orbit the Sun once?  Show how this calculation would be done.

9.  What is the period of Earth's orbit around the Sun?

10.  What is the size of Earth's orbit (in AU)?

11.  When you stand on the Earth's surface, you experience your "normal" Earth weight.  What would happen to your Earth weight if you were one Earth radius above the surface?  (That's twice as far from the center as simply standing on the surface.)

12.  What does gravitational force between 2 objects depend on?

Some questions from Newton's laws:

13.  A 10-kg object is pushed on by a 200-N force.  What will be the acceleration?

14.  What is the weight of a 100-kg man?

15.  Would the answer to 3 be different if he was on the moon?  How so?

16.  Consider yourself standing on a scale in an elevator.  The scale reads your weight.  Compared to being at rest, how would the scale reading change (if at all) if the elevator were:

A.  Moving with constant velocity upward

B.  moving with constant velocity downward

C.  Moving with constant acceleration upward

D.  Moving with constant acceleration downward

E.  If the cable snapped (yikes!) and the elevator were falling

(Answers below.)

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1.  Explain the meaning of "inverse square law".

The force (of gravity, in this case) gets progressively weaker by the factor 1 over the distance squared.  Double the distance --> force is 1/4 as great as it was.  Triple the distance --> force is 1/9 the original.

2.  Discuss each of Kepler's 3 laws.

See notes.

3.  At what point in its orbit is the Earth closest to the Sun?

Perihelion, which is approximately January 3-4 each year.

4.  At what point in its orbit is the Earth moving fastest?

Same point as 3 above.

5.  What causes seasons?

Tilt of Earth's axis.

6.  What is a semi-major axis of orbit (a)?

Half the longest distance across the orbital path (ellipse).

7.  What is an Astronomical Unit (AU)?

Defined as the semi-major axis of Earth's orbit - roughly 93,000,000 miles - or  half the longest width across Earth's orbit.

8.  Consider Jupiter.  It's orbit is 5 AU in size (roughly).  How long should it take Jupiter to orbit the Sun once?  Show how this calculation would be done.

5^3 = T^2

So, T = the square root of 125, or around 11 years.

9.  What is the period of Earth's orbit around the Sun?

1 year, or approximately 365.25 days.

10.  What is the size of Earth's orbit (in AU)?

Defined as 1 AU.

11.  When you stand on the Earth's surface, you experience your "normal" Earth weight.  What would happen to your Earth weight if you were one Earth radius above the surface?  (That's twice as far from the center as simply standing on the surface.)

1/4 your surface weight.

12.  What does gravitational force between 2 objects depend on?

mass of the objects; distance between; a universal (unchanging) constant (G)

13.  F = m a

200 = 10 a

a = 20 m/s/s

14.  W = m g

W = 100 g = 980 newtons

15.  Yes.  The weight would be smaller (1/6 as much, since Moon surface gravity is 1/6 that of Earth).

16.  a.   your regular weight
b.  your regular weight
c.  greater than your regular weight
d.  less than your regular weight
e.  zero!  (meaning that you are "weightless")

Exam 1 topics

General topics for exam 1.  Be sure to review all assigned homework, blog posts and your notes.

You are permitted to have a sheet of notes for this test.  I will NOT give equations.

SI units (m, kg, s) - meanings, definitions
velocity
average vs. instantaneous velocity
acceleration
related motion problems using the formulas
speed of light (c)
gravitational acceleration (g)
freefall problems
Newton's 3 laws - applications and problems
Kepler's 3 laws - applications and problems
epicycles
Newton's law of universal gravitation (inverse square law)
weight vs. mass

center of mass/gravity (to be covered on Tuesday)

Newton's take on gravity and orbits

Universal Gravitation (1687, Principia)

Newton's take on orbits was quite different. For him, Kepler's laws were a manifestation of the bigger "truth" of universal gravitation. That is:

All bodies have gravity unto them. Not just the Earth and Sun and planets, but ALL bodies (including YOU). Of course, the gravity for all of these is not equal. Far from it. The force of gravity can be summarized in an equation:

or.... the force of gravitation is equal to a constant ("big G") times the product of the masses, divided by the distance between them (between their centers, to be precise) squared.

Big G = 6.67 x 10^-11, which is a tiny number - therefore, you need BIG masses to see appreciable gravitational forces.

This is an INVERSE SQUARE law, meaning that:

- if the distance between the bodies is doubled, the force becomes 1/4 of its original value
- if the distance is tripled, the force becomes 1/9 the original amount
- etc.

Weight

Weight is a result of local gravitation. Since F = G m1 m2 / d^2, and the force of gravity (weight) is equal to m g, we can come up with a simple expression for local gravity (g):

g = G m(planet) / d^2

Likewise, this is an inverse square law. The further you are from the surface of the Earth, the weaker the gravitational acceleration. With normal altitudes, the value for g goes down only slightly, but it's enough for the air to become thinner (and for you to notice it immediately!).

Note that d is the distance from the CENTER of the Earth - this is the Earth's radius, if you're standing on the surface.

If you were above the surface of the earth an amount equal to the radius of the Earth, thereby doubling your distance from the center of the Earth, the value of g would be 1/4 of 9.8 m/s/s. If you were 2 Earth radii above the surface, the value of g would be 1/9 of 9.8 m/s/s.

The value of g also depends on the mass of the planet. The Moon is 1/4 the diameter of the Earth and about 1/81 its mass. You can check this but, this gives the Moon a g value of around 1.7 m/s/s. For Jupiter, it's around 2.5 m/s/s.

How things orbit

Nicolaus Copernicus, 1473 - 1543

http://astro.unl.edu/naap/ssm/animations/configurationsSimulator.html

Galileo Galilei, 1564 - 1642

http://galileo.rice.edu/

Galileo and his telescope:

moon craters
moons of Jupiter
sunspots
phases of Venus
"rings" of Saturn
stars in the Milky Way

Johannes Kepler, 1571-1630

Kepler's laws of planetary motion

http://astro.unl.edu/naap/pos/animations/kepler.swf



Note that these laws apply equally well to all orbiting bodies (moons, satellites, comets, etc.)

1. Planets take elliptical orbits, with the Sun at one focus. (If we were talking about satellites, the central gravitating body, such as the Earth, would be at one focus.) Nothing is at the other focus. Recall that a circle is the special case of the ellipse, wherein the two focal points are coincident. Some bodies, such as the Moon, take nearly circular orbits - that is, the eccentricity is very small.

2. The Area Law. Planets "sweep out" equal areas in equal times. See the applets for pictorial clarification. This means that in any 30 day period, a planet will sweep out a sector of space - the area of this sector is the same, regardless of the 30 day period. A major result of this is that the planet travels fastest when near the Sun.

3. The Harmonic Law. Consider the semi-major axis of a planet's orbit around the Sun - that's half the longest diameter of its orbit. This distance (a) is proportional to the period of time (P) to go around the Sun in a very peculiar fashion:

a^3 = P^2

That is to say, the semi-major axis CUBED (to the third power) is equal to the period (time) SQUARED. This assumes that we choose convenient units:

- the unit of a is the Astronomical Unit (AU), equal to the semi-major axis of Earth's orbit (approximately the average distance between Earth and Sun). This is around 150 million km or around 93 million miles

- the unit of time is the (Earth) year

The image below sums this up nicely (I hope):

Example problem:  Consider an asteroid with a semi-major axis of orbit (a) of 4 AU. We can quickly calculate that its period (P) of orbit is 8 years (since 4 cubed equals 8 squared).

Likewise for Pluto: a = 40 AU.  P works out to be around 250 years.

Note that for the equation to be an equality, the units MUST be AU and Earth years.


Cool:
http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/kepler6.htm

Newton problems and so forth

homework for Newton's laws, etc.

1. Describe each of Newton's 3 laws.

2. A 0.5 kg toy car is pushed with a 40 newton force. What is the car's acceleration?

3. Without calculating anything, what would be the effect (in problem 2) of increasing the mass of the car?

4. Give an example of Newton's 1st law in action.

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

6. Newton's "big book", what I claim is the most important non-religious book of all time is _____ and was published in _____.

7.  What are epicycles and why are they important in the history of science?

8. Distinguish between weight and mass.

9.   What is the SI unit of force?  What is the English unit of force?

10.  How does weight depend on gravitational acceleration?

11.   Freefall review.  Consider a ball falling from rest.  How fast would it be moving after 4 seconds?  How far would it fall in this time?

Newton and the science that came before him.

Brief history

Ancient science highlights:

Epicycles

Precession

From class this evening:

http://astro.unl.edu/naap/ssm/animations/ptolemaic.swf

http://astro.unl.edu/naap/motion3/animations/sunmotions.swf


The most important things to get out of this were:

- Epicycles were a very useful way to (wrongly) explain why retrograde motion happened with planets.

- Precession (the wobbling of the Earth) causes us to have different North Stars (or no North Star) at various points over the course of thousands of years.  Thus, star maps are not accurate after several hundred years.  However, this was not understood until the time of Newton and others.

Scientific Revolution

N. Copernicus, d. 1543

  De Revolutionibus Orbium Celestium

Galileo Galilei, 1564-1642

  Siderius Nuncius

  Dialogue on Two World Systems

(J. Kepler, C. Huygens, R. Descartes, et. al.)

Isaac Newton, 1642-1727

  Principia Mathematica, 1687

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.

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And now, in more contemporary language:

1.  Newton's First Law (inertia)

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

The 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 that force is unbalanced (greater than any resisting force), the object will accelerate.

Symbolically:

F = m a

That's a linear relationship.

Greater F means greater a.  However, if the force is constant, but the mass in increased, the resulting acceleration will be less:

a = F / m

That's an inverse relationship.

We have a NEW unit for force.  Since force = mass x acceleration, the units are:

kg m / s^2

which we define as a newton (N).  It's about 0.22 lb.

There is a special type of force that is important to mention now - the force due purely to gravity.  It is called Weight.  Since F = m a, and a is the acceleration due to gravity (or g):

W = m g

Note that this implies that:  weight can change, depending on the value of the gravitational acceleration.  That is, being near the surface of the Earth (where g is approximately 9.8 m/s/s) will give you a particular weight value, the one you are most used to.  However, at higher altitudes, your weight will be slightly less.  And on the Moon, where g is 1/6 that of the Earth's surface, your weight will be 1/6 that of Earth.  For example, if you weight 180 pounds on Earth, you'll weight 30 pounds on the Moon!


3.  Newton's Third 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.  Strange, isn't it?

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 (m and M, let's say) experience the same force:

m A = M a

That's a little trick to convey in letters but, the larger object (M) will experience the smaller acceleration (a), while the smaller object (M) experiences the larger acceleration (A).

Sites from class tonight

https://www.facebook.com/okgo/videos/10153210535420683/
OK GO video

https://www.youtube.com/watch?v=E43-CfukEgs
vacuum chamber / Brian Cox

Motion problems etc.

HW problems in motion

Woo Hoo – it’s physics problems and questions! OH YEAH!!

You will likely be able to do many of these problems, but possibly not all. Fret not, physics phriends! Try them all.

1. Determine the average speed of your own trip to school: in miles per hour. Use GoogleMaps or something similar to get the distance, and try to recall the time from your last trip. Use your trip from home to Towson, or something that makes sense to you. If possible, do it in miles per hour AND m/s.

2. Consider an echo-y canyon. You stand 200-m from the canyon wall. How long does it take the echo of your scream (“Arghhhh! Curse you Physics!!!”) to return to your ears, if the speed of sound is 340 m/s? (Sound travels at a constant speed in a given environment.) Also, keep in mind that the sound has to travel away from AND back to the source.

3. What is the difference between traveling at an average speed of 65 mph for one hour and a constant speed of 65 mph for one hour? Will you go further in either case?

4. What is the meaning of instantaneous velocity? How might we measure it?

5.  How far will a light pulse (say, a cell phone radio wave) travel in 1 second? In one minute? In one year? You don't have to work this out, but you should show HOW it would be calculated. Keep in mind that the light pulse travels AT the speed of light.

6.  What is the acceleration of a toy car, moving from rest to 6 m/s in 4 seconds?

7. What does a negative acceleration indicate?

8. Consider an automobile starting from rest. It attains a speed of 30 m/s in 8 seconds. What is the car’s acceleration during this period?

9. In the above problem, how far has it traveled in the 8 seconds?  This problem may be a little tricky. If you're feeling ambitious, feel free to read ahead in the blog - note that I didn't cover all of the material in tonight's class.

10. Review these ideas. Write down answers, if it would be helpful.

a. standards for the m, kg, and s. Know the original meaning of the standard, and the current standard (approximate meaning - don't worry about the crazy numbers)

11.  What exactly is gravitational acceleration and what is the significance of 9.8 m/s/s?

12.  A ball is dropped from rest from a great height. After 3 seconds, how fast is it traveling? How far did it fall in this time?  (You may use an approximate value of 10 m/s/s for g, and assume that there is no air resistance.)

13.  Revisit problem 12.  If this has been done on the Moon at the same height, would it take more, less or the same time to hit the ground?  How about on Jupiter?

14.  What are the SI standard units for length, time, and mass, and what are they currently based on?

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Answers:

1.  Answers will vary for this problem.  I will not ask you to convert to m/s on a test, but the conversion factor is approximately:  1 m/s = 2.25 mile/hour.

2.  v = d/t

340 = 2(200)/t

t = 1.2 sec

3.  no difference

4.  velocity at a given "instant", though there is no easy way to define an "instant."  Essentially, we mean the velocity at a practical "instant," say, at a particular second.

5.  Since v = d/t, d = v t.

So, the distance that light will travel will be equal to the speed of light (3 x 10^8 m/s) multiplied by the appropriate number of seconds.

In 1 second, d = 3 x 10^8 meters.
In 1 minute, d = (3 x 10^8) x 60 meters.
In 1 year, d = (3 x 10^8) x 60 x 60 x 24 x 365.35 meters.

6.  a = (vf - vi)/t

a = (6-0)/4 = 1.5 m/s/s

7.  "slowing down" relative to the direction you think of as positive.  For example, driving forward and hitting the brakes.

8.  a = (30 - 0)/8 = 3.75 m/s/s

9.  d = 0.5(vi + vf) t = 0.5(0 + 30) x 8 = 120 meters

10, 11.  see notes.

12.  Approximating g at 10 m/s/s:  v = 30 m/s.  d = 45 m

13.  :-)

14.  See notes

Gravity? Gravity!

Tonight we discuss the acceleration due to gravity - technically, "local gravity". It has a symbol (g), and it is approximately equal to 9.8 m/s/s, near the surface of the Earth. At higher altitudes, it becomes lower - a related phenomenon is that the air pressure becomes less (since the air molecules are less tightly constrained), and it becomes harder to breathe at higher altitudes (unless you're used to it). Also, the boiling point of water becomes lower - if you've ever read the "high altitude" directions for cooking Mac n Cheese, you might remember that you have to cook the noodles longer (since the temperature of the boiling water is lower).

On the Moon, which is a smaller body (1/4 Earth radius, 1/81 Earth mass), the acceleration at the Moon's surface is roughly 1/6 of a g (or around 1.7 m/s/s). On Jupiter, which is substantially bigger than Earth, the acceleration due to gravity is around 2.2 times that of Earth. All of these things can be calculated without ever having to visit those bodies - isn't that neat?

Consider the meaning of g = 9.8 m/s/s. After 1 second of freefall (falling without resistance - which is not exactly the case here but...), a ball would achieve a speed of .....

9.8 m/s

After 2 seconds....

19.6 m/s

After 3 seconds....

29.4 m/s

We can calculate the speed by rearranging the acceleration equation:

vf = vi + at

In this case, vf is the speed at some time, a is 9.8 m/s/s, and t is the time in question. Note that the initial velocity is 0 m/s.  In fact, when initial velocity is 0, the expression is really simple:

vf = g t

Got it?

The distance is a bit trickier to figure. This formula is useful - it comes from combining the definitions of average speed and acceleration.

d = vi t + 0.5 at^2

Since the initial velocity is 0, this formula becomes a bit easier:

d = 0.5 at^2

Or....

d = 0.5 gt^2

Or.....

d = 4.9 t^2

(if you're near the surface of the Earth, where g = 9.8 m/s/s)

This is close enough to 10 to approximate, so:

d = 5 t^2

So, after 1 second, a freely falling body has fallen:

d = 5 m

After 2 seconds....

d = 20 m

After 3 seconds....

d = 45 m

After 4 seconds...

d = 80 m

This relationship is worth exploring. Look at the numbers for successive seconds of freefall:

0 m

5 m

20 m

45 m

80 m

125 m

180 m

If an object is accelerating down an inclined plane, the distances will follow a similar pattern - they will still be proportional to the time squared. Galileo noticed this. Being a musician, he placed bells at specific distances on an inclined plane - a ball would hit the bells. If the bells were equally spaced, he (and you) would hear successively quickly "dings" by the bells. However, if the bells were located at distances that were progressively greater (as predicted by the above equation, wherein the distance is proportional to the time squared), one would hear equally spaced 'dings."

Check this out:

Equally spaced bells:

http://www.youtube.com/watch?v=06hdPR1lfKg&feature=related

Bells spaced according to the distance formula:

http://www.youtube.com/watch?v=totpfvtbzi0

Furthermore, look at the numbers again:

0 m

5 m

20 m

45 m

80 m

125 m

180 m

Each number is divisible by 5:

0

1

4

9

16

25

36

All perfect squares, which Galileo noticed - this holds true on an inclined plane as well, and its easier to see with the naked eye (and time with a "water clock.")

Look at the differences between successive numbers:

1

3

5

7

9

All odd numbers. Neat, eh?

FYI:

http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion1.html

Velocity and Acceleration

Mathematics of motion

Today, we are going to talk about how we think about speed and the rate of change in speed (usually called acceleration).  It is a bit math-y, but don't panic - we'll summarize things nicely in a couple of simple-to-use equations.

First, let's look at some definitions.

Average (or constant) velocity, v

v = d / t

That is, distance divided by time.  The SI units are meters per second (m/s).

* Strictly speaking, we are talking about speed, unless the distance is a straight-line and the direction is also specified (in which case "velocity" is the appropriate word).  However, we'll often use the words speed and velocity interchangeably if the motion is all in one direction (1D).

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)


What about.....
The Speed of light

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).

By the way, it's hard to remember this exact number, and I wouldn't expect you to.  However, here are some approximations that may make it easier to keep it in mind.  The speed of light is approximately:

- 300,000,000 meters/sec

- 186,000 miles/sec

- 7 times around the Earth's equator in 1 second

- Out to the Moon in around 1 second (1.3 seconds is closer) - so, the Moon is approximately 1.3 "light seconds" away (on average)

- To the Sun in about 8 minutes - so, the Sun is approximately 8 "light-minutes" away (on average)

- To Mars in about 13 minutes, though this varies depending on the relative locations of Earth and Mars in their respective orbits

- To the nearest (non-Sun) star, actually a 3-star system (Alpha Centauri A and B, and Proxima Centauri) in 4.3 years.  Yes, YEARS.  So, that 3-star system is around 4.3 "light years" away from us.  And that's our closest neighbors!!  See why we don't get too far in space travel?

Instantaneous Velocity

Average velocity should be distinguished from instantaneous velocity (what you get, more or less, 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 the idea of velocity is pretty useful if you care about the velocity at a specific time OR the average velocity for a trip. However, 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

Note that the i and f are subscripts.  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!).

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The equations of motion

Recall v = d/t.  That's usually how we calculate average velocity.  However, there is another way to compute average velocity:

v = (vi + vf) / 2

where vi is the initial velocity, and vf is the final (or current) velocity.  This is the same as taking the average of two numbers, in this case, the initial and final velocities.

Knowing these equations for average velocity, as well as the definition for acceleration, allows you to relate (or calculate) the interesting things about an object's motion:  initial velocity, final velocity, displacement, acceleration, and time.

As it happens, you can do a bunch of algebra to put the equations together into more convenient forms.  I will do this for you and summarize the most useful equations.

Today we will chat about the equations of motion. There are 4 very useful expressions that relate the variables in questions:

vi - initial velocity
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



Note that in each of the 4 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.

By the way, there is a 5th equation of motion  (d = vf t - 0.5 at^2) that is sometimes useful.  We won't need it in this class.)


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.

Also, if the initial velocity is zero (as it usually is in our problems), the equations become even simpler:

vf = at

d = 0.5 (vf) t

d = 0.5 at^2

Please keep THESE final 3 at your fingertips.


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:

Part 1

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

To start this problem, ask yourself:  "What do I know in this problem, and how can I represent these things as symbols?"

For example, "starting from rest" indicates that the initial velocity (vi) is zero.    "7 seconds" is the time, and "1.5 m/s^2" is the acceleration.

See if you can solve the problem from here.

Part 2

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!

Physics - YAY!