Physics exam 1
Example: $2.21 + 8% tax = $2.3868, rounds to
$2.39
Example: $2.21 + $1.35
$3.56
pictures of the 2 graphs in phone: 2-2 average speed and velocity
-2 graphs are the same -velocity = slope Δx/Δt graph in x-t graph
Notes on acceleration
-Acceleration is any change in velocity. This means acceleration not only refers to speeding up and slowing down, backwards or forwards, but also change in direction. -It is convenient to define a unit of acceleration 1 g = 9.8 m/s2. "g" is the magnitude of acceleration of a falling object near the surface of the earth.
what does a physicist do?
-Observes something (observes nature and tries to explain it) (ex: car moves = Newton's law -Builds a model of the observation (or equation) -Tests the model to gain insight on the world
throwing a ball
-accelerates -slows down to top -accelerates coming back down
ball dropping
-increase in acceleration until hitting the ground -velocity increasing until hitting the ground
dimensionally consistent
-unit of displacement x is meters, (m) -unit of velocity, v, is meters per second (m/s) -unit of time, t, is seconds (s) -unit of acceleration, a, is meters per second squared (m/s^2)
Water going over Angel Falls, in Venezuela, the world's highest waterfall, drops through a distance of 3212 ft What is this distance in km?
0.9790 1 foot = 0.0003048 km
ex: 1,000 g = ? kg
1 kg
ex: 1 kg = ? mg
1,000,000 mg
ex: 1 Gg = ? g
1,000,000,000 g
The x and y components of a vector r⃗ are rx = 13 m and ry = -8.5 m, respectively. 1.) Find the direction of the vector r⃗ 2.) Find the magnitude of the vector r⃗ 3.) Suppose that rx and ry are doubled. Find the direction and the magnitude of the new vector r⃗ ′.
1.) -33 degrees 2.) 16 3.) -33º and 31 m
Suppose that each component of a certain vector is doubled. 1.) By what multiplicative factor does the magnitude of the vector change? 2.) By what multiplicative factor does the direction angle of the vector change?
1.) 2 2.) 1
Vector A⃗ has a magnitude of 50 units and points in the positive x direction. A second vector, B⃗ , has a magnitude of 120 units and points at an angle of 70 ∘ below the x axis. 1.) Which vector has the greater x component. 2.) Which vector has the greater y component?
1.) A⃗ 2.) A⃗
2 ways to approach adding vectors
1.) Graphically line the two vectors head to tail and then measure the result. 2.) Break the vectors into components and add the components separately. 1.) graphing/drawing 2.) decompose the vectors/look at the vectors components
Here we will make 2 explicit assumptions about our study of motion:
1.) The motion of the object is along a straight line -Vertical, horizontal, or somewhere in between, just straight 2.) The object is a particle -Real objects are big. Mass is spread out over the volume; there is some surface area that can interact with things. For now we will assume everything is a point.
A man out walking his dog makes one complete pass around a perfectly square city block. 1.) Which of the following vectors is equal to r⃗ AB? 2.) Which of the following vectors is equal to −r⃗ AB? 3.) Which of the following vectors is equal to r⃗ AB−r⃗ DA?
1.) none of them (Recall that, for vectors to be equal, they must have the same magnitude and direction.) 2.) r⃗ CD only 3.) −(r⃗ CD+r⃗ DA) , r⃗ AB+r⃗ BC , and r⃗ BC−r⃗ CD
You are driving up a long, inclined road. After 1.50 mi you notice that signs along the roadside indicate that your elevation has increased by 500 ft 1.) What is the angle of the road above the horizontal? 2.) How far do you have to drive to gain an additional 160 ft of elevation?
1.) θ = sin^-1 (500/77920) = 3.6º 2.) sin(3.6) = 660/7920 + x --> 7920 + x = 660/sin(3.6) --> )660/sin(3.6)) - 7920 = x --> x = 0.49
Example of significant figures: A tortoise travels at 2.51 cm/s for 12.23 s. How far does the tortoise go?
2.51 cm/s × 12.23 s = 30.7 cm (three significant figures) -seconds cancel out -use 3 significant figures for the answer because 2.51 has 3 significant figures and you always use the smallest amount
1 inch = ? cm
2.54 cm
1 yard = ? feet
3 feet
The speed of light to five significant figures is 2.9979 × 10^8m/s. What is the speed of light to three significant figures?
3.00 x 10^8
clicker question: you are adding vectors of length 20 and 40 units. What is the only possible resultant magnitude that you can obtain out of the following choices? 0 18 37 64 100
37 you need to find the wrong ones: cant be zero because not opposite and the same length; longest possible length = 60 so it can't be above that; shortest possible length is 20 (40-20) so it can't be shorter than that (can't do 20-40 because you can't have a negative length)
clicker question: a certain vector has x and y components that are equal in magnitude. Which of the following is a possible angle for this vector in a standard x-y coordinate system?
45 degrees 45,45,90 triangle
ex: man travels 50 m in 8 seconds in the positive x direction, what is his velocity?
50/8 = 6.25 m/s
1 mile = ? feet
5280 feet
picture in phone: 2-3 instantaneous velocity
??
The Pythagorean theorem
H^2 = O^2 + A^2 ex: what is H? --> H = square root (O^2 + A^2)
slide 5: displacement vs distance
If you drive from your house to the grocery store and back, you have covered a distance of 8.6 mi, but displacement is zero. If you drive from your house to the grocery store and then to your friend's house, your displacement is -2.1 mi x and the distance you have traveled is 10.7 mi.
Adding vectors in different directions
In multiple dimensions, we still follow the same rule, line the vectors head to tail. Then we need to bring in our trig skills. -perpendicular arrows example: ^ | ---------------> A arrow going right and B arrow going up, the resultant vector would connect diagonally going up and to the right resultant = square root (A^2 + B^2) and θ = tan^-1 (B/A) R, A, and B are the magnitudes of the vectors
Problem Solving Tactics II: Brainstorming
Linking concepts with equations -write what is given after drawing a picture -identify what is known, what is unknown, what concept to apply, and what equation to use
Trigonometry
Many pictures that you will need to draw will involve making use of trigonometry, the study of angles. -sin -cos -tan SOH CAH TOA
does the displacement of an object depend on the specific location of the origin of the coordinate system?
NO -because the displacement is the difference between 2 coordinates, the origin does not matter
If the position of a car is zero, does its speed have to be zero?
NO -the speed does not depend on position; it depends on the change of position. Because we know that the displacement does not depend on the origin of the coordinate system, an object can easily start at x = -3 and be moving by the time it gets to x = 0
Instantaneous velocity & speed
Say that we can sample position at very small intervals of time. If we could let the time interval get infinitely small, we approach the instantaneous velocity. v = lim (Δx/Δt) = dx/dt lim is Δt --> 0 Even though the denominator is getting small, so is the numerator, so the fraction can still be large.
Adding vectors by component
The component vectors, if added head to tail, give the original vector. If we add the component vectors of two vectors, we then add the two vectors. slide 27: -(a) A⃗ + B⃗ = vector C; red lines = how they got vector A; blue lines = how they got vector B -(b) so... A's y coordinates + B's y coordinates = C's y coordinates (and same goes for C's x coordinates) -Ax is parallel to Bx --> Ax + Bx = Cx -Ay is parallel to By --> Ay + By = Cy -A⃗ means vector A; while A means length of A -vector C = vector Cx + vector Cy --> C = square root (Cx^2 + Cy^2)
Round-off error with significant figures
The last digit in a calculated number may vary depending on how it is calculated, due to rounding off of insignificant digits -5 and above = round up -4 and below = round down
instantaneous acceleration
The limit of a rate as the denominator approaches zero is called a derivative. Instantaneous acceleration is then the limit of average acceleration as the time interval approaches zero — or alternatively, acceleration is the derivative of velocity
Subtracting vectors
To subtract two vectors, we change the direction of the vector, add the two from tail to head. A better way to think about it is to draw the picture as the addition of two vectors. -ex: R = A - B *or* R + B = A -put the tips of the arrows together when subtracting -negative of a vector = opposite direction and negative length (ex: slide 28: if B is --> then -B would be <--)
Unit Vectors
Unit vectors are dimensionless vectors of unit length. -length equal to 1, all have the same length -adding 2 unit vectors doesn't get a unit vector, because 1 + 1 ≠ 1 (unlike regular vectors where adding 2 vectors get you a vector)
Decomposing vectors
We can separate the vector into components in both the x-direction and the y-direction using sines and cosines slide 26: -interested in vector A sooo the components are Ax and Ay -can decompose A⃗ into Ax and Ay -Ax is the horizontal component -Ay is the vertical component -length of A = square root (Ax^2 + Ay^2)
Scalars
We have already discussed in detail that every quantity in physics has both a quantity and a dimension. For anything that this completely describes, we can call this a scalar. Examples of scalars: -temperature -mass -height
Converting units Example: Vesna Vulovic survived the longest fall on record without a parachute when her plane exploded and she fell 6 miles, 551 yards. What is the distance in meters?
We have mixed units, so we first convert to a single unit that we can then convert to meters. 1 mile = 5280 ft and 1 yard = 3 ft The distance in feet is equal to: 6 mile (5280 ft / 1 mile) + 551 yards (3 ft / 1 yard) To convert from feet to meters: 1 meter = 3.281 ft (3.28083... ft) 33,333 ft * (1 meter / 3.281 ft) = 1.02 x 10^4 m 3 sig fig for answer because of 5280
Vectors
We live in a multi-dimensional world, some quantities manifest themselves in multiple dimensions. If a quantity has both a magnitude and a direction, we call this quantity a vector. -picture on slide: vector plotted on a graph, has a length and a measure angle θ -need number/magnitude, units, and direction -know length and direction
Adding vectors
We will often need to add multiple vectors together to find a result. If they both point in the same direction, it's easy. -Line the vectors head to tail and add the length of the two vectors together. ------------>-------> = --------------------> end resulting arrow = "resultant" (sum of 2 arrows)
chapter 2
Welcome to kinematics
Instantaneous speed
When the speed of an object is constantly changing, the instantaneous speed is the speed of an object at a particular moment (instant) in time
Problem Solving Tactics I: Drawing
Whenever possible, draw a picture. Physics is about observing the world and creating a model. Drawing pictures helps you take the words of a problem and envision what is happening. The pictures need not be works of art, they are to help you brainstorm.
Can an object accelerate if its speed is constant?
Yes, if its direction changes! average acceleration = Δv / Δt Average acceleration is in the direction of the change in velocity; i.e., towards the center of the circle
The time T required for one complete oscillation of a mass m on a spring of force constant k is T = 2pi square root (m/k) Find the dimensions k must have for this equation to be dimen-sionally correct.
[k] = kg/s^2
speed is
a scalar quantity -There is no direction associated with speed. -Define the average speed (always positive) as: average speed = distance / change in time
so velocity =
acceleration x time
cosθ =
adjacent angle / hypotenuse
cotθ =
adjacent angle / opposite angle
hypotenuse
always across from the right angle of the triangle
acceleration is
any change in velocity. This means acceleration not only refers to speeding up and slowing down, backwards or forwards, but also change in direction. -It is convenient to define a unit of acceleration 1 g = 9.8 m/s2. -"g" is the magnitude of acceleration of a falling object near the surface of the earth.
ex: a man travels backwards 50 m in 40 seconds, find the velocity when moving in the x (horizontal) direction
average velocity = -50 / 40 -1.25 m/s
ex: a man travels 50 m in 8 seconds, the the velocity when moving in the x (horizontal) direction
average velocity = 50 / 8 6.25 m/s Xi = 0 Xf = 50
calculate and predict motion example: how far do you get with a constant velocity of v = 50mi/h in 4 hours?
average velocity = change in position / change in time v = x/t x = vt
10^-2
centi- c
θ = cos^-1 (?)
cos^-1 (adjacent/hypotenuse)
10^-1
deci- d
Before describing motion, you must set up a coordinate system
define an origin and a positive direction.
We can define the displacement vector of the system as:
delta X = Xfinal - Xinitial
The magnitude of this vector is
delta x > 0 delta x < 0 delta x = 0
average velocity =
displacement / change in time (elapsed time) can be positive, zero, or negative velocity is a vector
does the odometer in a car measure distance or displacement?
distance
example: An observer, whose eyes are 1.83m above the ground, is standing 32.0 m away from the tree. The ground is level, and the tree is growing perpendicular to it. The observer's line of sight with the treetop makes an angle of 20.0o above the horizontal. How tall is the tree?
draw a picture: the base is 32.0 m, the angle is 20.0º. the observer's view is 1.83 m off the ground. We have an adjacent and want the opposite, so we should use the tangent: tan (20.0o) = h' / 32.0 m, h' = 32.0 m x tan(20.0o) = 11.64 m, but this is the height from the person's eyes! htree = hobserver + h' = 11.64 m + 1.83 m = 13.48 m = *13.5 m*
Referring to the vectors in the figure, express the sum A⃗ +B⃗ +C⃗ in unit vector notation
find the x and y coordinates to each one then add them (pay attention to directions) (2.1, 0.74)
Inverse trig functions
finds angle θ sin^-1 cos^-1 tan^-1
10^9
giga- G
If we take the limit at the time interval gets small, we get the
instantaneous acceleration. a = lim (Δv / Δt) = dv/dt = (d/dt)(dx/dt) = d^2x/dt^2 lim is Δt --> 0
Displacement
is the change in the position state of the system -a vector, has magnitude and direction = how far you've moved -i.e. final - initial.
clicker question: if each component of a vector is doubled, what happens to the angle of that vector?
it does not change
10^3
kilo- k
Distance
length away in respect to positions -how much ground you have covered -total length travel; positive
10^6
mega- M
10^-6
micro- μ
10^-3
milli- m
if velocity is negative
moving in the opposite direction
10^-9
nano- n
moving backwards =
negative velocity because negative position
tanθ =
opposite angle / adjacent angle
sinθ =
opposite angle / hypotenuse
θ = sin^-1 (?)
sin^-1 (opposite/hypotenuse)
graphing vectors
slide 25
does the speedometer in a car measure velocity or speed?
speed -clearly measures speed and not velocity. Velocity is a vector (depends on direction), but the speedometer does not care what direction you are traveling. It only measures the magnitude of the velocity, which is the speed
θ = tan^-1 (?)
tan^-1 (opposite/adjacent)
The change in velocity is
the acceleration. The average acceleration is given as: av = vfinal - vinitial / tfinal - tinitial OR Δv / Δt
Significant Figures
the number of digits in a quantity that are known with certainty -number of significant figures after multiplication or division is the number of significant figures in the least-known quantity -how many figures/digits that are needed for proper calculation -ex: calculator gives you 1.2358 m, but you only need 2 significant figures --> 1.2 m -need to use the same number of significant figures when using equations -*Always keep the smallest number* (ex: 2.51 has 3 numbers and 12.23 has 4 numbers, so the final answer should have 3 digits)
Kinematics is
the study of motion -study of how objects move -a description of motion. -*not* dynamics, motion caused by something We are going to start by examining motion in one spatial dimension and then move into multiple dimensions later.
clicker question: Given that A⃗ +B⃗ = C⃗ and that |A| + |B| = |C| how are vectors A and B oriented with respect to each other?
they are parallel to each other and in the same direction
clicker question: Given that A⃗ +B⃗ = C⃗ and that |A|^2 + |B|^2 = |C|^2 how are vectors A and B oriented with respect to each other?
they are perpendicular to each other this is a right triangle |A| means length of A
clicker question: if 2 vectors are given such that A⃗ +B⃗ = 0, what can you say about the magnitude and direction of vectors A and B?
they will have the same magnitude, but must be in opposite directions
Leading or trailing zeroes can make it hard to determine number of significant figures: like in 2500 and 0.000036 Each of these has how many significant figures?
two significant figures -in this case we should use scientific notation
Instantaneous velocity
velocity of an object in motion at a specific point in time. This is determined similarly to average velocity, but we narrow the period of time so that it approaches zero. If an object has a standard velocity over a period of time, its average and instantaneous velocities may be the same
so distance =
velocity x time because [L] = [L]/[T] multiply the right side by time ([T]) so that time cancels out so that [L] = [L] meaning distance = velocity x time
Position
where you are, your place, in respect to your surroundings
picture in phone of 6 graphs: labeled ABCDother and 123
x/t have constant = what is velocity? graph 1 = velocity is zero (D) graph 2 = velocity constant graph 3 = negative velocity A = 2 B = 3 C = acceleration
