Practice Problems

1. Consider a 0.1-kg coconut, falling from a tree 4-m above the ground. Find the following:

a. type (and amount) of energy before it falls
b. type (and amount) of energy right before it hits the ground
c. velocity right before it hits the ground
d. velocity at a point 1-m above the ground

2. Two toy cars collide on a track. A red car (0.2-kg) is traveling at 4 m/s and it smacks into the back end of a blue car (0.4 kg) traveling at 2 m/s in the same direction. They stick together. Find:

a. the type of collision that this is
b. the velocity of the red/blue cars (stuck together) immediately after the collision
c. what happens to the kinetic energy of the system before and after the collision. Calculate it, if you're not sure

3. Explain the following:

a. conservation of energy
b. conservation of momentum
c. elastic vs. inelastic collisions
d. the difference between mass and weight (and how to calculate it)

4. Two cars spring (explode) apart. If the first (1-kg) moves to the left at a final speed of 3 m/s, what is the speed of the second car (0.5-kg)?

5. Revisit the Newton's 2nd Law track problem: A 0.5-kg cart is pulled across a track by a string. The string is connected over a pulley to a 0.1-kg mass.

a. Find the acceleration of the cars
b. Find the acceleration of the cars if there is 0.5-N of friction between the car and track

6. How fast would a rollercoaster car have to travel so that you would just lose contact with the seat over a 5-m (radius) hill?

Session 10 - Energy

Session 9 - Newton's Second Law

As we have seen, Newton's second law is easy to express mathematically:

F = m a

Strictly speaking, the force (F) is the NET FORCE. The mass (m) refers to the mass of the system, and the acceleration (a) also refers to the system.

The unit is the kg m / s^2

This is defined as a newton (N).

We consider a few cases in detail.

1. Weight

First, the force due to gravity - weight, W

Since F = m a, and the acceleration is due to gravity,

W = m g

Mass (m) is the amount of matter (or stuff). The weight is a way of quantifying how much gravity pulls on the mass. This means, if g is different, the weight is also different. You weigh less on the Moon, more on Jupiter, less at high altitudes, etc.

2. Inclined Planes

Objects resting on inclined planes are not free to fall directly down. Rather, they are constrained by the geometry of the plane. Part of the weight (a parallel component) can be thought of as acting DOWN the plane. Part of the weight (a perpendicular component) can be thought of as acting ONTO the plane. By trig:

W(parallel) = mg sin(theta)

W(perpendicular) = mg cos(theta)

Without any resistance whatsoever, all objects slide down a plane with the same acceleration:

F = m a

W(parallel) = m a

mg sin(theta) = m a

a = g sin(theta)

3. Friction

Friction is a catch-all term for any resistive force (really due to electromagnetic interactions between surfaces).

The frictional force (f) is the amount of force that resists motion - that is, it acts in the direction opposite the motion. We can quantify friction by introducing a coefficient of friction (u).

u = f / mg

Meaning: the ratio of frictional force that exists to the weight of the object is defined as the coefficient of friction. Typically, this is a (unitless) number much less than 1.

4. Circular Motion

As Newton's 1st law would predict, any object that moves in a circular path has a REASON to do that - some force is causing it to happen. Recall that acceleration is a change in velocity - since velocity refers to a magnitude (speed) AND a direction, if the direction of a body is changing, it MUST be accelerating even if the speed does not change.

Consider a ball spinning on a string at a constant speed. Even if the speed remains constant, we know that the ball is accelerating - its direction is constantly changing. Some force must be causing that to happen. We call such a force - a center-directed force - centripetal. The center-directed acceleration that results is called centripetal acceleration (ac).

ac = v^2 / r

Or, to compute the magnitude of the centripetal acceleration, we take the speed squared and divide it by the radius of orbit.

The units of acceleration are still m/s^2.

Test practice problems, without answers

1. How high will a ball travel into the air, if thrown straight up at 25 m/s?

2. If this ball were thrown at a 40-degree angle (also at 25 m/s), find the following: time in air, horizontal displacement, max vertical displacement

3. Add the following vectors together, which are acting concurrently at a right angle to each other: 40 m/s and 60 m/s. Find the angle from the 40 m/s vector.

Session 7 part 3 - some historical info FYI

http://en.wikipedia.org/wiki/Isaac_Newton

This is really exhaustive - only for the truly interested.

This one is a bit easier to digest:

http://galileoandeinstein.physics.virginia.edu/lectures/newton.html

We'll return to Newton's gravitation (along with Kepler) later in the course.

For Galileo:

http://galileo.rice.edu/

http://galileo.rice.edu/bio/index.html

I also recommend "Galileo's Daughter" by Dava Sobel. Actually, anything she writes is pretty great historical reading. See also her "Longitude."

It is also worth reading about Copernicus and the Scientific Revolution.

For those of you interested in ancient science, David Lindberg's "Beginnings of Western Science" is amazing.

In general, John Gribbin's "The Scientists" is a good intro book about the history of science, in general. I recommend this for all bio and chem majors.

As a science major, you owe it to yourself to find out the history of your discipline. I think it will give you new perspective and respect.

Test 1 review book pages

Session 7 - Newton and his laws.

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.

Session 7 - some biographical details

Some background details will be discussed in class. Here are some dates of note:

Nicolaus Copernicus
1473 - 1543
De Revolutionibus Orbium Celestium


Tycho Brahe
1546 - 1601


Johannes Kepler
1571 - 1630
Astronomia Nova

Galileo Galilei
1564 - 1642
Siderius Nuncius
Dialogue on Two Chief World Systems
Discourse on Two New Sciences


Isaac Newton
1642 - 1727
Philosophiae Naturalis Principia Mathematica (1687)

Session 6 - more details on Projectiles

The Range Equation is very useful expression for determine the maximum horizontal displacement, when a projectile lands on the same plane from which it was launched.

R = (vo^2 / g) sin (2 theta)

This relationship (derived below) has an interesting prediction (since there is a sine of 2 times the angle). Since the sine relationship is symmetrical from 0 to 180 degrees (about the 90 degree point), the sine of any angle less than 90 degrees is the same as the sine of its supplementary angle. That is, the sine of 80 = the sine of 100.

Since there is a double angle, you get equal sine values for (twice) a launch angle and (twice) the compliment of that angle. For example, the sine of 40 = the sine of 50.

You could also think of this as: you get equal ranges for angle 45 + or - the same amount. The range is the same for 30 and 60, 10 and 80, 35 and 55, etc.

Pretty neat, eh?

Session 6 - Projectile Motion in general

Problem to try:

Consider a baseball hit at 40 m/s at an angle of 35 degrees. Find the following:

a. initial components of velocity

b. time to apogee

c. max height attained

d. max horizontal displacement

e. total time in air

f. the OTHER angle that would give the same horizontal displacement*

Session 5 - Projectile Motion

Projectile motion is a beautiful, but slightly tricky application of vectors. It's vector math mixed with one-dimensional motion.

First, let us consider vector components. Just like 2 vectors can be added together to give a resultant vector, a single vector can be "deconstructed" into its "perpendicular components."

If the angle is measured with respect to horizontal, a vector V at angle theta (T) has components:

Vh = V cos (T)

Vv = V sin (T)

Horizontal (h)

Vertical (v)

And now, projectiles:

Consider first a projectile launched with nothing but horizontal velocity (vox), from a vertical height (xy). There is NO horizontal acceleration, ONLY vertical acceleration. So, the ball lands in the same exact time as if it had been simply dropped. True, it goes further horizontally, but it doesn't take any longer to do that. This will be demonstrated in class.

To solve projectile motion problems, you still use equations of motion, but you use them with horizontal and vertical variables - and you NEVER mix the horizontals with verticals. Time (t) is neither horizontal nor vertical - it is the same (t) for both dimensions.

Given an initial horizontal velocity (vox) of 12 m/s and vertical height (xy) of 5 m, find:

a. time to fall (t)

b. horizontal displacement (xx)

Practice problems from the text

Hello,

By popular demand, I present you some suggested textbook problems from College Physics, 9th Edition, by Serway/Vuille.

* tricky problems

Trig review:

Chapter 1: 41-49 odd

One-dimensional Motion:

Chapter 2: 7, 29, 33, 37, 45, 53*

Vectors:

Chapter 3: 8, 10, 11

Session 4, part 3 -- Vector Addition yet again!

Adding vectors which are at NON-right angles is a bit trickier. Draw a parallelogram and determine the diagonal. You will need to use the law of cosines, but must first find the angle opposite the diagonal. Details furnished in class.

c^2 = a^2 + b^2 - 2ab cos (C)

where C is the angle across from c (the side). In practice, this angle is (again) the supplement of the angle between the vectors.

Session 4, part 2 -- Vector Addition

Vector quantities can be visualized as arrows. Adding two arrows in the same direction gives a larger arrow. Likewise, adding two (similar) vector quantities (say, 2 velocities) yields a larger velocity.

Subtracting vector quantities is simply adding one vector to a negative vector, resulting in a smaller vector.

Imagine walking on a "people mover". If you walk in the direction of the mover's velocity, you ultimately have a greater velocity than the mover itself (with respect to the people observing on the side). If you walk in the OPPOSITE direction to the mover's velocity, which is a pretty lame thing to do, you will have a velocity that is less than that of the mover itself.

How about adding vectors at angles? This is a bit trickier. See the second slide above.

If two vectors, A and B, are at a right angle to each other, the "resultant vector" C is given by the Pythagorean Theorem:

A^2 + B^2 = C^2

and the angle between C and the horizontal is given by the tangent relationship.

tan (angle) = OPP / ADJ = A/B

so that the angle itself is given by the inverse (or arc) tangent of A/B:

angle = arctan (A/B)

If trigonometry is not something you remember that well, check out:

http://howdoweknowthat.blogspot.com/2009/07/how-far-away-is-that.html


Some problems to try:

1. A flagpole makes an angle of 55 degrees with respect to horizontal, as measured from a point 10-m away. How tall is the flagpole?

2. Add 2 vectors together: 10 and 20, at a right angle to each other. Find the resultant vector and the angle as measured from the 20 vector.

3. Imagine trying to cross a river, 100-m wide with a current of 2 m/s. If you head out in your motorboat at 10 m/s, find the following:

a. how long it will take you to cross
b. how far downstream you will be
c. the angle you take with respect to your original path

In this problem, there is NO acceleration - rather, the velocities are added simultaneously (or "concurrently").

Session 4, part 1 -- Problems to try

Hiya. First a comment from yesterday's class. I remarked that that the "area under the curve" for a v vs. t graph gives displacement. This is true - see the image from yesterday's class (in the previous blog post). The units of (v times t) gives units of displacement. For those of you who speak calculus, we refer to this area as an integral, a process made easier if you have a mathematical function that describes the velocity as a function of time.

If this makes no sense to you - fret not, physics friends. Simply know that the area under the curve gives you displacement.

And now, some problems to play with:

Woo Hoo – it’s physics problems with motion! OH YEAH!!!

1. Determine the average velocity of your own trip to school: in m/s and miles per hour.

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!!!”), if the speed of sound is 340 m/s? (Sound travels at a constant speed.)

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 can we measure it?

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

6. How far will a light pulse (say, a cell phone radio wave) travel in 10 minutes?

7. What does a negative acceleration indicate? How about a negative velocity? Negative displacement?

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, and how far has it traveled?

9. You are driving down a dark road when out pops a deer, 40 m away. Your speed is 20 m/s and you instantly slam on the brakes providing your car with a -2 m/s2 acceleration. Will you hit the deer? (Hint: Find out how far your car will travel given this acceleration, assuming that you come to a complete stop.)

10. What is the acceleration due to gravity? What does this value mean?

11. How does the acceleration due to gravity vary on the Moon? On Jupiter?

12. If you are “pulling 5 g’s”, what acceleration do you experience?

13. How long will it take a rock falling from rest to drop from a 100-m cliff?

14. If you drop a pebble from a bridge into a river below, and it takes 2.5 seconds to hit water, how high is the bridge?

15. Drop a bowling ball from atop a high platform. How fast will it be traveling after 3 seconds of freefall?

16. In order to pass another car on the highway, you increase your velocity from 25 m/s to 30 m/s. This occurs in 10 seconds. What is your acceleration? How far have you moved during this time?

17. You kick a soccer ball straight up into the air with an initial velocity of 22 m/s. How long will it take to reach apogee? To return to the Earth? How high will it rise? What will be its final velocity before hitting Earth?

18. Draw two (approximate) graphs of motion (displacement vs. time, velocity vs. time) for each of the following scenarios:
a. A car accelerates from rest for 5 seconds, travels at a constant velocity for 5 seconds, and then slows down for 5 seconds, finally stopping for 5 seconds.
b. A person walks at a constant velocity for 10 seconds, stops for 5 seconds, then walks back to where they started (faster) for 5 seconds.
c. You toss a ball up in the air and it lands back in your hand.

Session 3 (one more time!) - Graphs of Motion

An object's motion can be represented on graphs:

x vs. t

v vs. t

a vs. t

For a non-moving object, all 3 graphs resemble horizontal lines.

For an object moving with constant velocity:

the x vs. t graph is linear - that is, the displacement increases by the same amount each time interval.

For an object moving with constant acceleration, the displacement increases exponentially with respect to time.

See image above.

The slope of an x vs. t graph will give the velocity. Of course, if the graph is a curve, things are a bit complicated. You could take a tangent line to a particular point on the curve - then take the slope of the tangent line to get the instantaneous velocity at that point.

The slope of a v vs. t graph will give the acceleration.

Session 3 redux - The acceleration due to gravity

Tonight we're discussing the acceleration due to the ubiquitous force of gravity.

Contrary to popular belief, all objects do NOT experience the same force of gravity. An object experiences a gravitation force directly related to its mass. That is, greater mass equals greater gravitational force. All objects undergo the same acceleration (in the absence of air resistance) for a quite subtle reason. (Short answer: there is a greater attractive force of gravity for more massive objects; however, the greater force of attraction is offset by the greater inertia and resistance to motion. The two effects exactly cancel each other.)

Near the surface of the earth, the acceleration due to gravity (g) is:

g = 9.8 m/s^2

Which is to say, an object will accelerate (in free-fall) at a rate of 9.8 m/s^2 - increasing its speed by 9.8 m/s with each second of free-fall. Or, if it is rising, losing 9.8 m/s with each second of upward motion.

So, after 1 second of free-fall, the object will attain a speed of 9.8 m/s.

After 2 seconds, 19.6 m/s.

After 3 seconds, 29.4 m/s.

And so on.

Galileo's odd numbers rule was also discussed in class. Galileo determined that accelerating objects fall (or roll) through total displacements that are proportional to the amount of time squared. If we approximate g as 10 m/s^2, then:

x(1 sec) = 5 m

x(2 sec) = 20 m

x(3 sec) = 45 m

x(4 sec) = 80 m

Note that these displacements are proportional to 1, 4, 9, 16 - perfect squares. And interestingly, the differences between these "base" displacements are 1, 3, 5, 7...... odd numbers. Neat, eh?

Sessions 2 and 3

Hi again, and welcome back to Physics 101! So nice to see you again. How've you been?

First, let us chat about order of magnitude questions, so-called "Fermi Questions" (as they were made legendary by Enrico Fermi). Here are a few we will discuss in class:

Number of days to walk around the world
Number of times your blink in your lifetime
Number of revolutions a car's tire will make
Number of dollar bills to stack to the Moon
Number of hairs on your head
Area needed for all people on Earth to gather for a party

We will discuss strategies for solving these questions today. Now clearly these are not physics questions, per se, but the way of thinking about them can be informative to our process. Physics is a way of thinking about problems, and often applying the language of mathematics to weird situations.

OK? OK!!!

And now......

THE EQUATIONS OF MOTION!

First, let's look at some definitions.

Vector vs. Scalar:

Vector quantities - quantities that must be expressed with a magnitude AND direction. Examples: velocity, displacement, force, momentum, magnetic field, electric field

Scalar quantities - quantities that only requires a magnitude (mass, time, energy, speed)

>

Average velocity

v = x / t

That is, displacement divided by time.

Another way to compute average velocity:

v = (v0 + v) / 2

where vo ("v naught") is the original velocity, and v is the final (or current) velocity.

Average velocity should be distinguished from instantaneous velocity:

v(inst) = x / t, where t is a very, very, very tiny time interval.

[In calculus speak, this is: v(inst) = dx/dt, a derivative.]

>

Acceleration, a

a = (change in velocity) / time

a = (v - vo) / t

The units here are m/s^2.

>

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

vo - "v naught", or original velocity. Note that the 0 is a subscript.
v - velocity at a specific point (usually at time t)
a - acceleration
t - time
x - displacement

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

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

v (avg) = x / t

v (avg) = (vo + v) / 2

and the definition of acceleration:

a = (change in v) / t or

a = (v - vo) / t

Through the miracle of algebra, these can be manipulated (details shown in class) to come up with:

v = v0 + at

x = 0.5 (vo + v) t

x = vo t + 0.5at^2

v^2 = vo^2 + 2ax

x = v t - 0.5at^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.

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

Finally, find:

- the total distance the car travels during this "experiment"
- graphs of motion for: displacement vs. time, velocity vs. time, acceleration vs. time

Got it? Hurray!

Physics - YEAH!!!

Session 1

Howdy, and welcome to Physics 101!

Some comments on the first class. I speak about SI units at some length. To remind you:

Mass is measured based on a kilogram (kg) standard.
Length (or displacement or position) is based on a meter (m) standard.
Time is based on a second (s) standard.

How do we get these standards?

Length - meter (m)

- originally 1 ten-millionth the distance from north pole (of Earth) to equator
- then a distance between two fine lines engraved on a platinum-iridium bar
- (1960): 1,650,763.73 wavelengths of a particular orange-red light emitted by atoms of Kr-86 in a gas discharge tube
- (1983, current standard): the length of path traveled by light during a time interval of 1/299,792,458 seconds

That is, the speed of light is 299,792,458 m/s. This is the fastest speed that exists. Why this is is quite a subtle thing. Short answer: the only things that can travel that fast aren't "things" at all, but rather massless electromagnetic radiation. Low-mass things (particles) can travel in excess of 99% the speed of light.

Long answer: See relativity.

Time - second (s)

- Originally, the time for a pendulum (1-m long) to swing from one side of path to other
- Later, a fraction of mean solar day
- (1967): the time taken by 9,192,631,770 vibrations of a specific wavelength of light emitted by a cesium-133 atom

Mass - kilogram (kg)

- originally based on the mass of a cubic decimeter of water
- standard of mass is now the platinum-iridium cylinder kept at the International Bureau of Weights and Measures near Paris
- secondary standards are based on this
- 1 u (atomic mass unit, or AMU) = 1.6605402 x 10^-27 kg
- so, the Carbon-12 atom is 12 u in mass

Volume - liter (l)

- volume occupied by a mass of 1 kg of pure water at certain conditions
- 1.000028 decimeters cubed
- ml is approximately 1 cc

Temperature - kelvin (K)

- 1/273.16 of the thermodynamic temperature of the triple point of water (1 K = 1 degree C)
- degrees C + 273.15
- 0 K = absolute zero

For further reading:

http://en.wikipedia.org/wiki/SI_units

http://en.wikipedia.org/wiki/Metric_system#History

>

In addition, we spoke about the spherocity of the Earth and how we know its size. I've written about this previously. Please see the blog entries below:

http://howdoweknowthat.blogspot.com/2009/07/how-do-we-know-that-earth-is-spherical.html

http://howdoweknowthat.blogspot.com/2009/07/so-how-big-is-earth.html


Physics - Yeah!!!