Imagine that you are throwing a baseball off a cliff, but doing so straight up (while standing at the edge of the cliff). The ball will go straight up and come straight back down alongside the cliff, landing at at the base of the cliff 30 m below. The initial upward velocity of the baseball is 25 m/s.
a. How long will it take the baseball to reach its maximum height?
b. How high up will it travel?
c. How long would it take to come right back to your hand (where it left)?
d. How long (in total) would it take to leave your hand, go up and travel to the very bottom of the cliff? There is a long way to solve this problem, but also a shorter (one equation) way: Note that the shorter method will require the use of the quadratic formula. If you don't like using the quadratic formula, you can find a solver online:
http://www.math.com/students/calculators/source/quadratic.htm
e. Finally, what is the approximate shape of the graphs of d vs. t and v vs. t?
Wednesday, September 28, 2016
Monday, September 26, 2016
Practice problem for quiz (which is Wednesday A and Thursday E) - answers below
On the quiz (Wednesday for A, Thursday for E), you will answer questions related to: unit conversions, Fermi approximation questions, equations of motion (which will be given) and graphs of motion. Review the previous homework and your notes.
Practice problem:
This problem assumes that the acceleration is that of near-Earth surface gravity, 9.8 m/s/s.
Consider an ball dropped from rest, now in freefall with minimal air resistance It is easiest to define DOWN as positive.
a. What is the ball's speed after 2.5 seconds of fall?
b. How far has the ball fallen in this time?
c. Draw an approximate graph of motion for d vs. t.
d. Draw an approximate graph of motion for v vs. t.
e. If, after 2.5 seconds of fall. the ball hits a swimming pool that is 2-m deep with water. Imagine that the ball slows down uniformly during its descent to the bottom finally hitting a speed of 4 m/s right before it hits bottom. What is the ball's acceleration during this descent through water, and how long did it take to reach the bottom?
Answers:
a. Vf = Vi + at = 0 + (9.8)2.5 = 24.5 m/s
b. d = Vi t + 0.5 at^2 = 0 + (0.5)(9.8)(2.5^2) = 30.6 m
c. basic parabola
d. linear
e. Use the 4th equation of motion for the easiest solution.
4^2 = 24.5^2 + 2 a (2)
a = -146 m/s/s
Yes, that's a pretty big deceleration, so maybe not totally realistic.
Using the first equation of motion:
Vf = Vi + at
4 = 24.5 + (-146)t
t = 0.14 sec
Practice problem:
This problem assumes that the acceleration is that of near-Earth surface gravity, 9.8 m/s/s.
Consider an ball dropped from rest, now in freefall with minimal air resistance It is easiest to define DOWN as positive.
a. What is the ball's speed after 2.5 seconds of fall?
b. How far has the ball fallen in this time?
c. Draw an approximate graph of motion for d vs. t.
d. Draw an approximate graph of motion for v vs. t.
e. If, after 2.5 seconds of fall. the ball hits a swimming pool that is 2-m deep with water. Imagine that the ball slows down uniformly during its descent to the bottom finally hitting a speed of 4 m/s right before it hits bottom. What is the ball's acceleration during this descent through water, and how long did it take to reach the bottom?
Answers:
a. Vf = Vi + at = 0 + (9.8)2.5 = 24.5 m/s
b. d = Vi t + 0.5 at^2 = 0 + (0.5)(9.8)(2.5^2) = 30.6 m
c. basic parabola
d. linear
e. Use the 4th equation of motion for the easiest solution.
4^2 = 24.5^2 + 2 a (2)
a = -146 m/s/s
Yes, that's a pretty big deceleration, so maybe not totally realistic.
Using the first equation of motion:
Vf = Vi + at
4 = 24.5 + (-146)t
t = 0.14 sec
Thursday, September 22, 2016
Extra HW problem for A block (Monday) and E (Tuesday)
Don't forget about the others posted earlier.
Consider a rail gun launching a small projectile at high speeds. The projectile is launched from rest and accelerates at 200 m/s/s for 0.5 seconds.
What is the speed after this brief acceleration?
How far will the projectile have traveled?
If the projectile enters a magnetic braking zone (right after it reached the speed just attained) and is brought to a stop in 2 seconds, what is the acceleration during this time?
Draw approximate graphs that relate d vs t and v vs t for this entire problem.
Wednesday, September 21, 2016
Some notes on motion - read at your leisure in the next few days. Note that there is homework below this post.
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).
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) -
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?
Some velocities to ponder....
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.
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!).
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!).
>
The equations of motion
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: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.
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
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:
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!
Homework! (Not to be turned in.) For Thursday (A, if you have time) and Friday (E)
1. 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.
2. 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?
3. What is the meaning of instantaneous velocity? How might we measure it?
4. 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.
5. What is the acceleration of a toy car, moving from rest to 6 m/s in 4 seconds?
6. What does a negative acceleration indicate?
7. 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? Also, how far has it traveled in the 8 seconds?
2. 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?
3. What is the meaning of instantaneous velocity? How might we measure it?
4. 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.
5. What is the acceleration of a toy car, moving from rest to 6 m/s in 4 seconds?
6. What does a negative acceleration indicate?
7. 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? Also, how far has it traveled in the 8 seconds?
8. Review these ideas. Write down answers, if it would be helpful. 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)
Monday, September 19, 2016
HW reminder
Hey everyone -
In class, you took data for a toy car. Come to class with a graph of distance (y-axis) vs. time (x-axis). Make sure to get the slope of the graph.
If at all possible, use a computer graphing program - Logger Pro is my preferred program (and you'll need to download it, using the recently sent link). Others will work, as long as you remember to generate a line and slope.
Also, calculate the "instantaneous" speed through the photogate: width of index card divided by average time.
Look at your two values (photogate method vs. ticker tape slope) - which do you trust more? How close are they? Be prepared to discuss this in class.
Thanks!
sean
In class, you took data for a toy car. Come to class with a graph of distance (y-axis) vs. time (x-axis). Make sure to get the slope of the graph.
If at all possible, use a computer graphing program - Logger Pro is my preferred program (and you'll need to download it, using the recently sent link). Others will work, as long as you remember to generate a line and slope.
Also, calculate the "instantaneous" speed through the photogate: width of index card divided by average time.
Look at your two values (photogate method vs. ticker tape slope) - which do you trust more? How close are they? Be prepared to discuss this in class.
Thanks!
sean
Monday, September 12, 2016
HW to submit on Wednesday (A) and Thursday (E)
Show all of your work, please. Write out your work, or if you work on a computer, print it out.
1. How many toilets are there in all of the major league football stadiums, baseball parks, and basketball arenas in the USA?
2. Create a conversion factor to convert from (1) inches per second to AU (Astronomical Unit) per moment. A moment is a medieval unit of time equal to 1.5 minutes (or 1/40 of an hour), and an AU is used to represent astronomical distances (and is half the longest length of Earth's orbitequal to 1.5 x 10^11 m).
3. How many times will you blink in your lifetime?
Also, find out this information: what are the dimensions for the trip from your house to school? Give the distance and the time. This is not part of your homework to turn in, but rather a place to start our next class discussion. Feel free to use Google Maps, or the data from an actual car trip.
1. How many toilets are there in all of the major league football stadiums, baseball parks, and basketball arenas in the USA?
2. Create a conversion factor to convert from (1) inches per second to AU (Astronomical Unit) per moment. A moment is a medieval unit of time equal to 1.5 minutes (or 1/40 of an hour), and an AU is used to represent astronomical distances (and is half the longest length of Earth's orbitequal to 1.5 x 10^11 m).
3. How many times will you blink in your lifetime?
Also, find out this information: what are the dimensions for the trip from your house to school? Give the distance and the time. This is not part of your homework to turn in, but rather a place to start our next class discussion. Feel free to use Google Maps, or the data from an actual car trip.
Thursday, September 8, 2016
HW for Monday (A) and Tuesday (E)
Here are some unit problems to play with. You will NOT turn them in - rather, we will review them in class.
1. Create a factor to convert from m/s to light-years (LY) per millennium. 1 LY = 9.4607 x 10^12 km.
2. Create your own conversion factor - m/s to something else interesting.
3. How long is a micro-century? Give your answer in units that are appropriate (seconds? days?). Also, remember that micro equals 1 millionth.
1. Create a factor to convert from m/s to light-years (LY) per millennium. 1 LY = 9.4607 x 10^12 km.
2. Create your own conversion factor - m/s to something else interesting.
3. How long is a micro-century? Give your answer in units that are appropriate (seconds? days?). Also, remember that micro equals 1 millionth.
Now, some Fermi questions to play with: calculate answers. Resist the urge to look up answers or techniques.
1. How many hairs are on your head?
2. How many times does one of the tires on a car rotate in its lifetime (before the tire is changed)?
3. If ALL of the people in the world gathered together for the world's biggest party, and they were all to meet at the same location, how big of an area would be needed? As big as a state? Which state? Bigger? A country? Try to calculate/estimate this.
1. How many hairs are on your head?
2. How many times does one of the tires on a car rotate in its lifetime (before the tire is changed)?
3. If ALL of the people in the world gathered together for the world's biggest party, and they were all to meet at the same location, how big of an area would be needed? As big as a state? Which state? Bigger? A country? Try to calculate/estimate this.
Wednesday, September 7, 2016
HW for Thursday (A) if you see this in time. HW for Friday (E).
Answer these in your notes. This is posted very late for A block, so do it if you have time. E block - this is due Friday.
Find out the meaning of these prefixes:
centi
milli
micro
nano
pico
femto
atto
kilo
mega
giga
tera
peta
Also, show how you would convert these quantities. Do it the long way, though you can use an online converter to check your answers if you like.
10 m to ___ ft
5.5 inches to ___km
14 seconds to ___hr
3 dimes to ___dollars
Find out the meaning of these prefixes:
centi
milli
micro
nano
pico
femto
atto
kilo
mega
giga
tera
peta
Also, show how you would convert these quantities. Do it the long way, though you can use an online converter to check your answers if you like.
10 m to ___ ft
5.5 inches to ___km
14 seconds to ___hr
3 dimes to ___dollars
Tuesday, September 6, 2016
Thursday, September 1, 2016
HW for Tuesday (A) and Wednesday (E)
Today we started discussing standards.
1. The SI meter was originally based on the distance between the earth's North Pole and its equator. How was the size of the earth originally determined?
2. Why do YOU suppose the standard of the meter was eventually changed?
3. Create your standard for anything - describe it. Example - the kardashian is (unofficially) defined as 72 days of marriage.
SI Units!
Some comments on standards. We generally use SI units in physics. To inform 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
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