Physics exam 1
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
T/F If A + B =0, then the vectors A and B have equal magnitudes and are directed in the same direction.
False
T/F If A +B = C and A^2+B^2=C^2, then the vectors A and B are oriented perpendicular relative to one other.
True
temperature SI unit
kelvin (K)
10^3
kilo- k
mass SI unit
kilogram (kg)
Distance
length away in respect to positions -how much ground you have covered -total length travel; positive
Vector M = 4.00 m points eastward and vector N = 3.00 m points southward. The resultant vector M + N is given by
3^2 + 4^2 = x^2 x = 5 m cosθ = 4/5 θ = 36.9º 5.00 m at an angle 36.9 ° south of east.
so energy =
mass x (velocity)^2
10^6
mega- M
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
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.
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
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)
There is a special case and special solution if we determine the projectile begins and ends at the same height. We call this the "range equation"
*V0x = V0 cosθ* *V0y = V0 sinθ* x = V0x t y = V0y t - 1/2gt^2 t = x/V0x 0 = V0y t - 1/2gt^2 --> V0y = 1/2gt V0 sinθ = 1/2g(x/V0 cosθ) --> x = (V0^2/g) 2sinθcosθ = *(V0^2/g) sin2θ* = *R*
throwing a ball
-accelerates -slows down to top -accelerates coming back down
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)
ex: 1 Gg = ? g
1,000,000,000 g
Two youngsters dive off an overhang into a lake. Diver 1 drops straight down, and diver 2 runs off the cliff with an initial horizontal speed v0. 1.) Is the splashdown speed of diver 2 greater than, less than, or equal to the splashdown speed of diver 1? 2.) Choose the best explanation from among the following: -Both divers are in free fall, and hence they will have the same splashdown speed. -The diver who drops straight down gains more speed than the one who moves horizontally. -The divers have the same vertical speed at splashdown, but diver 2 has the greater horizontal speed.
1.) greater 2.) The divers have the same vertical speed at splashdown, but diver 2 has the greater horizontal speed.
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
You're driving at a constant velocity of 45 mph when you notice Big Foot in the middle of the road. You hit the breaks and deaccelerate at a rate of -5.2m/s^2. If Big Foot is 40m away, do you hit him?
45 mph = 20 m/s V^2 =V0^2 + 2ax --> 0 = 20^2 + 2(-5.2)(x) --> *38 m* NO YOU DONT
ex: man travels 50 m in 8 seconds in the positive x direction, what is his velocity?
50/8 = 6.25 m/s
A vector A has components Ax = 12.0 m and Ay = 5.00 m. What is the angle that vector A makes with the x-axis?
5^2 + 12^2 = X^2 X = 13 cosθ = 12/13 *θ = 22.6º*
Dimensional analysis
Any valid physical formula must be dimensionally consistent - each term must have the same dimensions
Many factors can influence the motion of a falling object.
Ex: Air resistance, latitude, large heights above the surface of earth.
We have discussed in great detail what happens in one dimension, either horizontal or vertical or in between. What happens in multiple dimensions?
First, all of our vectors become multi-dimensional. We have x, y, and sometimes z components of position, velocity, and occasionally acceleration.
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
When a football in a field goal attempt reaches its maximum height, how does its speed compare to its initial speed?
It is less than its initial speed.
Objects in free-fall
Near the surface of the earth, all objects fall subject to the same acceleration: gravity = 9.8 m/s2 (32.2 ft/s2).
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.
Scientific notation
Scientific notation writes these as a number from 1-10 multiplied by a power of 10, making the number of significant figures much clearer: 2500 = 2.5 × 10^3 or 0.000036 = 3.6 x 10-5 -each now has 2 significant figures
slide 11: A special case, 0 change in height
Snapshots of a trajectory; red dots are at t = 1 s, t = 2 s, and t = 3 s -we have velocity in y direction, general launch angle, start at zero -Note that maximum height: H = V0y^2/g -total flight time: Ttot = 2V0y/g -both determined by y-motion only! -total flight distance in x: range R -determined by x and y motion: R -y = height -x = range
Why not SI?
Some "non-standard" units are just too convenient. The minute is a better unit to measure the time of say, lectures. -ex: 50 minutes = 0.866 hours = 3000 s Some units are just traditional. The barn is a unit of area useful to nuclear scientists that is based off SI units -ex: 1 barn = 10-24 m2
slide 16: 4-5 Projectile Motion: Key Characteristics
Symmetry in projectile motion:
example: A goalie kicks a stationary 0.43 kg soccer ball sitting on a horizontal field. The ball travels a horizontal distance of 55 m and the ball lands at an angle of 43º from the horizontal. With what speed was the ball kicked?
The angle that the ball lands is the same as the angle that it was kicked. Also, the ball starts and stops at the same height. -We can use the range equation: x = (V0^2/g)sin(2θ) 55 = (V0^2/9.81) sin(43) V0 = 28.1 m/s
slide 15: 4-5 Projectile Motion: Key Characteristics
The range is a maximum when θ = 45°: *Rmax = V0^2/g* -constant velocity = find angle -average/middle height thrown will produce the farthest distance
Projectile Motion
The special case is projectile motion. A projectile is any moving object subject only to the force of gravity. ax = 0 and ay = -g (9.8 m/s2, down) Each motion continues as if the other motion is not present. -you can describe the vectors separately (x and y)
T/F Free fall is the motion of an object subject only to the influence of gravity.
True
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)
Equations of Kinematics: V^2 =
V^2 = (initial velocity)^2 + 2(acceleration)(position) V^2 = V0^2 + 2ax could also be written as V^2 (t) = V0^2 + 2a(V0t + 1/2at^2)
ex: projectile motion
Vx = V0 cos θ Vy = V0 sin θ tan θ = V0y/V0x
A ball is thrown with an initial speed of 60 m/s at an angle of 30° above the horizontal. What is the ball's horizontal displacement at the end of 4 seconds? Useg = 10 m/s2.
Vx = V0cosθ Vx = 60cos30 Vx = 52 x = V0x t x = 52(4) x = 208
equations for problems in x direction
Vx = V0x +ax(t) x = V0x (t) + 1/2(ax)(t)^2 Vx^2 = V0x^2 + 2(ax)(x)
Motion is symmetric
We are familiar with the expression what goes up, must come down. A better expression to help remember physics is as it goes up, so it goes down. Here's what I mean: When the object falls back to its original height, the speed of the object is the same as the speed when the object was released (the direction changed, so the velocity is different).
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
Why SI?
We use the metric system in science (and engineering) as an international standard. 1 m in the United States is 1 m in Europe is 1 m on the moon. The only unit conversions in SI are converting powers of 10. The fewer calculations you have to make, the less likely you are to cause a disaster. -easy to keep track, everything in powers of 10
Sorting out the vectors
We will use subscripts to keep tabs on our vectors. If a component is in the x-direction, we use the subscript, x (such as Vx). And, for the y-direction, we use the subscript, y (such as Vy). Our horizontal displacement equation and our vertical displacement equation look similar, can we describe motion in two dimensions using a set of kinematics equations in the x-direction and a second set in the y-direction?
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
Dimension of velocity
[L]/[T] aka velocity is measure in lengths over a period of time
Dimension of acceleration
[L]/[T^2]
Dimension of area
[L^2]
Dimension of volume
[L^3]
Dimension of energy
[M][L^2]/[T^2]
parabola
a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. The path of a projectile under the influence of gravity ideally follows a curve of this shape.
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
luminous intensity SI unit
candela (cd)
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*
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
distance SI unit
meter (m)
chapter 1
what is physics
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 <--)
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
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)
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.
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
current SI unit
ampere (A)
clicker question: if each component of a vector is doubled, what happens to the angle of that vector?
it does not change
Units can also help bail us out. When working problems, anytime you write a number,
keep the unit attached to it. If your units do not match the unit of the quantity you are calculating, something is not right!
10^-6
micro- μ
10^-3
milli- m
amount SI unit
mole (mol)
10^-9
nano- n
tanθ =
opposite angle / adjacent angle
sinθ =
opposite angle / hypotenuse
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
Position
where you are, your place, in respect to your surroundings
Example: $2.21 + 8% tax = $2.3868, rounds to
$2.39
ex: 1,000 g = ? kg
1 kg
1 yard = ? feet
3 feet
1 mile = ? feet
5280 feet
10^9
giga- G
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.
Example: $2.21 + $1.35
$3.56
average acceleration vector (ā)
(v - v0) / (t - t0) velocity - initial velocity / time - initial time
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
What are the basic elements that we need, to describe what we can measure? what are units used for?
-Time -Distance -Mass -Plus a few others we will need later -use units to solve a problem, units cancel out
time dependence of acceleration, velocity, and position (picture in phone)
-first graph: acceleration vs time graph shows a straight line, and a = gravity acceleration does not change/is constant (means velocity changes steadily) -second graph: velocity vs time graph shows a line going down v = v0 - gt velocity decreases to zero and then switches direction (speed decreases to zero, then increases)
picture in phone: 2-7 freely falling objects
-freely falling from rest means initial velocity = 0 -Δx increases with time -Δv does not change -acceleration = 9.81 -V0 = 0 -picture shows velocity at different times (0 seconds, 1 second, 2 seconds, etc.) -want distance traveled, use formula: x = V0t + 1/2at^2
ball dropping
-increase in acceleration until hitting the ground -velocity increasing until hitting the ground
picture in phone: trajectory of objects with initial velocity
-position increases = velocity increases -trajectory of a projectile: a lava bomb is thrown up in the air with an initial velocity of 29.4 m/s -position vs time graph shown -takes 3 seconds to go up and 3 seconds to come down -ex: at its highest point (3 seconds) x = V0t + 1/2at^2 x = 29.4(3) + 1/2(9.81)(3)^2 x = 0
example: A diver runs horizontally with a speed of 1.20 m/s off a platform that is 10.0 m above the water. What is his speed just before striking the water?
-there is no change in the x direction because there is no acceleration -y = zero because hitting the water -h = 10 -V0 = 1.20 Vy^2 = V0y^2 - 2g(y-h) Vy = square root (1.2^2 - 2(9.81)(0-10)) Vy = 14 m/s V^2 = Vy^2 + Vx^2 V = square root (1.2^2 + 14^2) V = 14.1 m/s
At the buzzer, a basketball player shoots a desperation shot. The ball goes in 1.)Determine the algebraic signs of the ball's x velocity and y velocity the instant after it leaves the player's hands. 2.) Determine the algebraic signs of the ball's x velocity and y velocity at the ball's maximum height.
1.) +, + 2.) +, 0
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
Two divers run horizontally off the edge of a low cliff. Diver 2 runs with twice the speed of diver 1 1.) When the divers hit the water, is the horizontal distance covered by diver 2 twice as much, four times as much, or equal to the horizontal distance covered by diver 1? 2.) Choose the best explanation from among the following: -The drop time is the same for both divers. -All divers in free fall cover the same distance. -Drop distance depends on t^2.
1.) twice as much 2.) The drop time is the same for both divers
Something that weighs 1 pound has a mass of about how many kg?
1/2 kg
1 inch = ? cm
2.54 cm
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
picture in phone: 2-3 instantaneous velocity
??
You go home to Pawnee, IN for spring break. You decide to visit your high-school friend, Ron, while you're there. In order to get to his house, you need to drive around Lil' Sebastian lake. A) What is the total distance you have traveled? B) What is the displacement vector D? C) What is the magnitude of the displacement vector? D) What is the angle of displacement from east? picture one slide 5 of recitation week 1
A.) 16 km B.) y = -3.21 km ; x = 8.21 km C.) 8.8 km D.) 21.4º
When you see a traffic light turn red, you apply the brakes until you come to a stop. If your initial speed was 17 m/s, and you were heading due west, what was your average velocity during braking? Assume constant deceleration.
Constant deceleration means that the average velocity (Vav) is just: average velocity = (initial velocity + final velocity) / 2 17 + 0 / 2 = 8.5
4-3 Zero Launch Angle example: (slide 7 pictures) a person jumping off of a cliff into water vs a person falling off a cliff into the water
Launch angle: direction of initial velocity with respect to horizontal -left picture = horizontal and vertical velocity in this case V0x = V0 cos θ V0y = V0 sin θ -
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
2D kinematics equations
Main idea: Horizontal and vertical motions are independent!
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
4-2 Projectile Motion: Basic Equations (slide 5 graphs)
The acceleration is independent of the direction of the velocity -when an object is falling vertically, then ax = 0 and ay = gravity -when an object is being thrown horizontally, up and away, ax = 0 and ay = gravity -only force is gravity (accelerating down) -in graph to the right (object thrown horizontally), velocity goes horizontally -we know that g down, often we choose +y up, then ay = -g
Chapter 4
Towards the "real" world
A crew on a spaceship saw a comet pass them. The crew wants to examine the comet and start to follow it immediately. If the comet is traveling a constant 2,500m/s and it takes the crew 2hr to catch it, what would the rockets acceleration be? (Assuming it had a constant acceleration, and the initial velocity of the rocket is 0)
V = V0 + at 2500 = 0 + (a)(3600) t = 0.69 m/s^2
A car is traveling at 30.0 m/s when the driver suddenly applies the brakes, giving the car a constant deceleration. The car comes to a stop in a distance of 120.0 m. What was the deceleration of the car?
V^2 = V0^2 + 2ay 0 = 30^2 + 2(a)(120) a = -3.75 deceleration = 3.75 m/s^2
A cork shoots out of a champagne bottle at an angle of 34.0º above the horizontal. If the cork travels a horizontal distance of 1.40 m in 1.30 s , what was its initial speed?
X = V0xt 1.4 = V0x (1.3) V0x = 1.08 V0x = V0 cos θ 1.08 = V0 cos(34) V0 = 1.3
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
Dimension of distance
[L] aka distance is measure in lengths
During the 2018 Winter Olympics, Sarah Hendrickson, a ski jumper, starts her first run at the top of a 187m hill. The angle of the hill is 37⁰ and the jumping ramp is at a height of 112m. What is her speed when she reaches the ramp assuming she starts at rest?
a = 9.8cos37 a = 5.9 m/s^2 sin37 = 75/x x = 124.6 V^2 = V0^2 +2ax V^2 = 0 + 2(5.9)(124.6) V = 38 m/s
As a train accelerates away from a station, it reaches a speed of 4.6 m/s in 5.4 s. If the train's acceleration remains constant, what is its speed after an additional 6.5 s has elapsed?
acceleration = 4.6/5.4 = 0.852 vfianl = vinitial + at 4.6 + (0.852)(6.5) = final velocity final velocity = 10.1
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
A pilot drops a bomb from a plane flying horizontally at a constant speed. Neglecting air resistance, when the bomb hits the ground the horizontal location of the plane will
be over the bomb
10^-2
centi- c
Equations of Kinematics
connect displacement, velocity, acceleration, & time Hints: 1.) Keep careful track of positive & negative directions 2.) Deceleration is same as negative acceleration 3.) With squares there can be positive & negative answers. Use common sense 4.) Demensional analysis
Person A and person B are both pushing a box across the floor. The box experiences a net force of 35 N at an angle of 52 deg. With how much force is person A pushing? Person B?
cos52 = y/35 y = 22 N 22^2 + x^2 = 35^2 x = 28 person A = 28 N person B = 22 N
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
Which of the following is a vector quantity? A) mass B) volume C) speed D) time E) displacement
displacement
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
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
picture example in phone: a drag racer starts from rest and accelerates at 7.40 m/s^2. How far has it traveled in (a) 1.00 second? (b) 2.00 seconds?
given: a = 7.4; V0 = 0; t = 1 or 2 seconds unknown: x use formula: x = V0t + 1/2at^2 x(1) = 3.7 m x(2) = 14.8 m
example: Dropping supplies A plane flying at a height of 1470 m must airdrop supplies to a town it is approaching. If the plane is flying at a constant speed of 300 m/s, how far from the target must the plane release the supplies? You can neglect air resistance.
height = 1470 V0x = 300 V0y = 0 x = ? y = h + V0yt - 1/2gt^2 0 = 1470 + (0)t - 1/2(9.81)(t)^2 t = 17.3 seconds x = V0xt x = (300)(17.3) x = 5,200 m
It is not always convenient to use the basic unit to report some quantities. We can resize the unit using
metric prefixes -These powers of 10 are what we call orders of magnitude!
if velocity is negative
moving in the opposite direction
The 2D kinematics variables (slide 3 table)
need an x and y component of everything -displacement (x and y) -acceleration (ay and ax) -final velocity (Vx and Vy) -initial velocity (V0x and V0y) -time (t)
moving backwards =
negative velocity because negative position
average velocity vector (v̅) =
rf - ri / t - t0 position final - position initial / time - time initial
The International System (SI)
the metric system
Before we get started, we need some essential building blocks so that we can communicate with other students, scientists, engineers, etc...
units
Equations of Kinematics: v =
v = initial velocity + (acceleration)(time) v = V0 + at v(t) = velocity at a specific time (could be written as v(t) = ...) -larger acceleration = larger velocity, and vise versa
A hockey puck slides off the edge of a table with an initial velocity of 20.0 m/s. The height of the table above the ground is 2.0 m. What is the magnitude of the vertical component of the velocity of the puck just before it hits the ground?
v^2 = v0^2 + 2ay V^2 = 0^2 + 2(9.81)(-2) V = 6.26 m/s
example: a ball is thrown vertically up from the ground with a speed of +5.00 m/s and later falls back to the ground What is the total time the ball remains in the air?
x = V0t + 1/2gt^2 x = 5 + 1/2(9.81)(t) answer: 1.02 seconds
Equations of Kinematics: x =
x = initial velocity (time) + 1/2(acceleration)(time)^2 x = V0t + 1/2at^2 x(t) = position at a specific time (could be written as x(t) = ....)
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
displacement =
Δr = r final - r initial
ex: 1 kg = ? mg
1,000,000 mg
At a given instant, the acceleration of a certain particle is zero. This means that
the velocity is not changing at that instant.
Jordan's Jump Michael Jordan's vertical leap is reported to be 48 inches. What is his takeoff speed? Give your answer in meters per second.
velocity^2 = (initial velocity)^2 + 2ax velocity^2 = (0)^2 + 2(9.8)(1.2192) velocity = square root (2(9.8)(1.2192)) velocity = 4.9
equations for problems in y direction
Vy = V0y +ay(t) y = V0y (t) + 1/2(ay)(t)^2 Vy^2 = V0y^2 + 2(ay)(y)
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
Distinction between velocity and acceleration vectors:
-velocity vector always points in direction of motion or is tangent to motion -acceleration can point in other directions other than direction of particle's motion
Consider 2 case: 1.) A ball is dropped, from rest, completely vertical off a 123m cliff. 2.) The same ball is thrown with an initial velocity of 35m/s in the horizontal direction off the same cliff. Find: a.) Total time for both b.) Final velocity of both c.) For case 2, how far away (horizontally) from the cliff did the ball land
1.) a.) y = V0t + (1/2)at^2 --> 123 = 0 + 1/2(9.81)t --> t = *5.0 sec* b.) V = V0 + at --> V = 0 + (9.81)(5) --> *V = 49 m/s* 2.) a.) x = V0t + (1/2)at^2 --> 123 = 0 + 1/2(9.81)t --> t = *5.0 sec* b.) Vy = 49 and Vx = 35 --> V^2 = Vx^2 + Vy^2 --> V^2 = 35^2 + 49^2 --> *V = 60 m/s* c.) x = vt --> x = (35)(5) --> *x = 175 m*
You are given the following vectors: A = (10, 4) B = (0, 8) C = (3, 5) Find and sketch 1.) A + B 2.) C - A 3.) A + 2B + C 4.) B - A - C
1.) (10, 12) 2.) (-7, 1) 3.) (13, 25) 4.) (-13, -1)
An arrow is shot at an angle of θ = 45º above the horizontal. The arrow hits a tree a horizontal distance D = 220m away, at the same height above the ground as it was shot. Use g = 9.8m/s^2 for the magnitude of the acceleration due to gravity. 1.) Find ta, the time that the arrow spends in the air. 2.) Suppose someone drops an apple from a vertical distance of 6.0 meters, directly above the point where the arrow hits the tree. How long after the arrow was shot should the apple be dropped, in order for the arrow to pierce the apple as the arrow hits the tree?
1.) (V0^2/g) sin2θ = R --> 220 = (V0^2/9.8) sin(2 x 45) --> 46.4 = V0 Vx = V0 cos θ --> Vx = 46.4 cos(45) --> Vx = 32.8 x = Vx t --> 220 = 32.8 t --> *t = 6.7* 2.) y = V0 t + 1/2at^2 --> 6 = 0t + 1/2(9.81)(t)^2 --> t = 1.1 6.7 - 1.1 = 5.6 --> *wait 5.6 seconds*
To celebrate a victory, a pitcher throws her glove straight upward with an initial speed of 5.3 m/s. 1.) How much time does it take for the glove to return to the pitcher? 2.) How much time does it take for the glove to reach its maximum height?
1.) 1.1 seconds (because 2 x (v/g)) 2.) half of time it took to reach the pitcher's hand --> 1.08/2 = 0.54 seconds
Playing shortstop, you pick up a ground ball and throw it to second base. The ball is thrown horizontally, with a speed of 25 m/s, directly toward point A (Figure 1). When the ball reaches the second baseman 0.52 s later, it is caught at point B 1.) how far is he from the second baseman? 2.) What is the distance of vertical drop, AB?
1.) 25 x 0.52 = 13 m 2.) Vy = V0y + ay(t) --> Vy = 0 + (9.81)(0.52) --> Vy = 5.1012 Vy^2 = V0y^2 + 2(ay)(y) --> (5.1012)^2 = 0 + 2(9.81)(y) --> y = 1.3
example: Football games often begin with a referee tossing a coin in the air to determine who kicks off. The referee tosses the coin with a speed of +5.00 m/s. 1.) How high does the coin go above the point of release? 2.) What is the total time the coin is in the air?
1.) V = V0 + at 5 = 0 + (9.81)(t) t = 0.510 seconds x = V0t + 1/2gt x = 0(0.510) + 1/2(9.81)(0.510) Ans: 1.28 m 2.) 0.510 x 2 (because it takes the same amount of time to get to the highest point as i does coming down) Ans: 1.02 s
While taking a Segway tour through Washington DC, Jamie loses her balance, leans forward, and speeds up at a constant acceleration. After 10 seconds, Jamie decides to jump off the Segway. Knowing their initial speed was 2 m/s with an acceleration of 3 m/s^2, find: 1.) Jamie's velocity before bailing 2.) Jamie's distance traveled from the start of their acceleration
1.) V = V0 + at --> V = 3 + (3)(10) --> V = 33 m/s 2.) x = V0t + (1/2)at^2 --> x = 2(10) + 1/2(3)(10)^2 --> 170 m
Tracy throws a ball straight up in the air with an initial velocity of 12.2m/s from a height of 2.0m. The ball reaches some maximum height and then she catches it at the initial height (2.0m). Assuming there is no air resistance and the only acceleration is from gravity, find: 1.) The maximum height 2.) Total time in the air 3.) Plot the velocity vs time graph
1.) V^2 =V0^2 + 2a(xmax) --> 0 = (12.2)^2 + 2(9.81)(x - 2) --> *x = 9.6 m* 2.) a = (V - V0) / t --> 9.81 = (0 - 12.2) / t --> t = 1.2 --> 1.2 x 2 = *2.4* 3.)
A woman stands at the edge of a cliff, holding one ball in each hand. At time t0, she throws one ball straight up with speed v0 and the other straight down, also with speed v0. 1.) If the ball that is thrown downward has an acceleration of magnitude a at the instant of its release (i.e., when there is no longer any force on the ball due to the woman's hand), what is the relationship between a and g, the magnitude of the acceleration of gravity? 2.) Which ball has the greater acceleration at the instant of release? 3.) Which ball has the greater speed at the instant of release? 4.) Which ball has the greater average speed during the 1-s interval after release (assuming neither hits the ground during that time)? 5.) Which ball hits the ground with greater speed?
1.) a=g 2.) Neither; the accelerations of both balls are the same. 3.) Neither; the speeds are the same. 4.) the ball thrown downward 5.) Neither; the balls hit the ground with the same speed.
The infamous chicken is dashing toward home plate with a speed of 6.0 m/s when he decides to hit the dirt. The chicken slides for 1.0 s, just reaching the plate as he stops (safe, of course). 1.) What is the magnitude of the chicken's acceleration? 2.) What is the direction of the chicken's acceleration? 3.) How far did the chicken slide?
1.) acceleration = (final velocity - initial velocity) / t --> (0-6) / 1 --> a = 6 m/s^2 2.) opposite the direction of motion 3.) x = V0t + 1/2at^2 --> x = 6(1) + 1/2(-6)(1)^2 --> x = 3.0
A battleship simultaneously fires two shells toward two identical enemy ships. One shell hits ship A, which is close by, and the other hits ship B, which is farther away. The two shells are fired at the same speed. Assume that air resistance is negligible and that the magnitude of the acceleration due to gravity is g. 1.) What shape is the trajectory (graph of y vs. x) of the shells? 2.) For two shells fired at the same speed which statement about the horizontal distance traveled is correct? 3.) Now, consider for the remaining parts of the question below that both shells are fired at an angle greater than 45 degrees with respect to the horizontal. Remember that enemy ship A is closer than enemy ship B. Which shell is fired at the larger angle? 4.) Which shell is launched with a greater vertical velocity, vy? 5.) Which shell is launched with a greater horizontal velocity, vx? 6.) Which shell reaches the greater maximum height? 7.) Which shell has the longest travel time (time elapsed between being fired and hitting the enemy ship)?
1.) parabola 2.) The shell fired at an angle closest to 45 degrees lands farther away 3.) A 4.) A 5.) B 6.) A 7.) A
A rocket blasts off and moves straight upward from the launch pad with constant acceleration. After 3.0 s the rocket is at a height of 87 m. 1.) What is the magnitude of the rocket's acceleration? 2.) What is the direction of the rocket's acceleration? 3.) What is its speed at this time?
1.) x = V0t + 1/2at^2 --> 87 = 0(3) + 1/2a(3)^2 --> a = 19.3 2.) upward 3.) V = V0 + at --> V = 0 + (19.3)(3) --> V = 58
A sailboat runs before the wind with a constant speed of 3.5 m/s in a direction 35º north of west. 1.)How far west has the sailboat traveled in 27 min ? 2.) How far north has the sailboat traveled in 27 min ?
1620 x 3.5 = x x = 5.7 1.) 5.7 x cos(35) = distance distance = 4.6 2.) 5.7 x sin(35) = distance distance = 3.3
The peregrine falcon is the fastest bird on earth with an airspeed of 242 mi/hr. What is this in m/s? Hint: 1 mi = 1609 meters
242 x 1609 = 389378 m 389378 m/s x 60 s/1 min x 60 min/ 1 hr = 1.40 x 10^9
A certain projectile is launched with an initial speed v0. At its highest point its speed is 1/2v0. What was the launch angle of the projectile?
At highest point, the vertical motion is momentarily zero, so the only component of speed at the apex is the horizontal component, which is constant throughout the flight assuming no air friction. So we know that vx = x0 cosθ = v0/2 so we know that cosθ = 1/2 thus θ = 60 degrees
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
slide 9: 4-3 Zero Launch Angle
This is the trajectory of a projectile launched horizontally -why shape of parabola, like y ~ x^2? -since x changes like t, x ~ t --> t ~ x -y shows parabola in time: y ~ t^2 and in space: y ~ x^2 -if i replace t with x in an equation = parabola? -because x is linear to t -Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In a previous atom we discussed what the various components of an object in projectile motion are. In this atom we will discuss the basic equations that go along with them in the special case in which the projectile initial positions are null (i.e. x0 = 0 and y0 = 0)
projectile motion equations
Vx = V0x X = V0xt Vx^2 = V0x^2 Vy = V0y - gt y = h + V0yt - 1/2gt^2 Vy^2 = V0y^2 - 2g(y-h) -h is height/initial position (start height) -velocity is a vector (direction throwing) -x is the distance horizontally/magnitude -y is the distance vertically/magnitude
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
The change in velocity is
the acceleration. The average acceleration is given as: av = vfinal - vinitial / tfinal - tinitial OR Δv / Δt
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
example: The Opening Kickoff The player kicked the ball 66.0 yards. The ball hung in the air for approximately 3.80 seconds. What was the velocity in m/s and angle of the opening kickoff? What was the maximum height of the kickoff? (1 yard = 3 ft; 3.281 ft = 1 m)
x = V0x x t 60.3504 = V0x x 3.8 V0x = 15.9 m/s velocity at max height: Vy = V0y - gt 0 = V0y -(9.81)(1.9) V0y = 18.6 m/s V^2 = Vy^2 + Vx^2 V = square root ((15.9)^2 + (18.6)^2) *V = 24.5 m/s* tan θ = V0y/V0x tan θ = 18.6/15.9 *θ = 49.5º* Vy^2 = Vy0^2 - 2gh (18.6)^2 = (24.5)^2 - 2(9.81)(h) *h = 17.7 m*
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
time SI unit
seconds (s) minutes hours
θ = sin^-1 (?)
sin^-1 (opposite/hypotenuse)
graphing vectors
slide 25
θ = tan^-1 (?)
tan^-1 (opposite/adjacent)
