LE 01-10b

Physics 231Topic 3: Vectors and two dimensional motionAlex BrownSeptember 14-18 2015

MSU Physics 231 Fall 2014#Whats up? (Monday Sept 14)

1) Homework set 01 due Tuesday Sept 15th 10 pm

2) Learning Resource Center (LRC) in 1248 BPS Monday 9 am to 1 pm and 3 pm to 5 pm Tuesday 9 am to 9 pm Friday 9am to 1 pm

MSU Physics 231 Fall 2014#Key Concepts: 2D MotionVectors and ScalarsTwo Dimensional MotionVelocity in 2DAcceleration in 2DProjectile motionThrowing a ball or cannon fireUniform Circular MotionCentripetal acceleration

Covers chapter 3 in Rex & Wolfson

MSU Physics 231 Fall 2014#You throw a ball straight up (positive direction) and catch it again at the same location where you released it. At the highest point the acceleration is:

Impossible to tell-9.8 m/s29.8 m/s20Extra Credit QuizMSU Physics 231 Fall 2014#SOH CAH - TOA:sin=opposite/hypotenuse =a/ccos=adjacent/hypotenuse=b/ctan=opposite/adjacent =a/b

Pythagorean theorem:Note that sin,cos,tan are dimensionless.

2 radians corresponds to 360o

abcTrigonometry

MSU Physics 231 Fall 2014#hyy = h sin() = h cos()

Always check carefully which angle is givenBe Careful.

yMSU Physics 231 Fall 2014# Scalar: A quantity specified by its magnitude only Vector: A quantity specified both by its magnitude and direction.

To distinguish a vector from a scalar quantity, it is usually written with an arrow above it, or in bold to distinguish it from a scalar.

Scalar: A Vector: A or A (bold face)Vectors and ScalarsMSU Physics 231 Fall 2014#Vectors and Scalars

MSU Physics 231 Fall 2014# Are these two vectors the same? Are the lengths of these two vectors the same?Two vectors are equal if both their lengths and directions are the same!QuestionNoYesMSU Physics 231 Fall 2014#ABA+BABB+AA+B=B+AVector AdditionMSU Physics 231 Fall 2014#Vector operations in equationsABA+Byx(xa,ya)(xb,yb)(xa+b,ya+b)

Example:A+BMSU Physics 231 Fall 2014#ABA-B = A + (-B) Vector Subtraction-B

MSU Physics 231 Fall 2014#A-BA-B=A+(-B)BVector Subtraction

MSU Physics 231 Fall 2014#abCartesian Coordinates:(x,y)=(a,b)

Also, r = a + b unit vector in x direction unit vector in y direction

rPlane Polar Coordinates(r,)

More on Coordinate Systems

xyMSU Physics 231 Fall 2014#Vector length and its componentsYx(xa,ya)Length of vector (use pythagorean theorem):

MSU Physics 231 Fall 2014#Vector operations in equations

MSU Physics 231 Fall 2014#Vector operations in equations

ffMSU Physics 231 Fall 2014#x2=2.5 cos()=1.77=-45oY2=2.5 sin()=-1.77QuestionA man walks 5 km/h. He travels 12 minutes to the east,30 minutes to the south-east and 36 minutes to the north.A) What is the displacement of the man when hes done?B) What is the total distance he walked?1 km2.5 km3 km

A)B) 1 + 2.5 + 3 = 6.5 kmMSU Physics 231 Fall 2014#Boat crossing the river

FlowMSU Physics 231 Fall 2014#QuestionA boat is trying to cross a 1-km wide river in the shortest way (straight across).Its maximum speed (in still water) is 10 km/h. The river is flowing (south) with 5 km/h.

1) At what angle does the captain have to steer the boat the gostraight across?

2) How long does it take for the boatto cross the river?

MSU Physics 231 Fall 2014#A boat is trying to cross a 1-km wide river in the shortest way (straight across).Its maximum speed (in still water) is 10 km/h. The river is flowing with 5 km/h.At what angle does the captain have to steer the boat the go straight across?

2) How long does it take the boat to cross the river?sin = opposite/hypotenuse = 5/10 = 0.5 = sin-1(0.5) = 30ovelocityhor=sqrt(100-25) = 8.66 km/h

Time = (1 km)/(8.66 km/h) = 0.115 h = 6.9 minFlow=5km/hv=10 km/hAnswer

MSU Physics 231 Fall 2014#3) If it doesnt matter at what pointthe boat reaches the other side, at what angle should the captain steerto cross in the fastest way?0o :the horizontal component of the velocity is then maximum.MSU Physics 231 Fall 2014#plane in the wind

MSU Physics 231 Fall 2014#Displacement in 2D

ffMSU Physics 231 Fall 2014#Velocity in 2D

MSU Physics 231 Fall 2014#2d motion; decompose intohorizontal and vertical components1d motion

MSU Physics 231 Fall 2014#Near the surface of the earth with the y-axis pointing up

ax = 0

ay = -g

So for motion in x-direction

and for motion in y direction the same equations(A-E) as before but using y (rather than x)

MSU Physics 231 Fall 2014#

(A)

(B)

(C)

(D)

(E)positive y points upso a = -gMSU Physics 231 Fall 2014#Pop and Drop

MSU Physics 231 Fall 2014#Pop and DropABFor A: vy = - gt2 vx = 0

y = y0- gt2 x = 0

For B: vy = - gt2 vx = v0

y = y0 gt2 x =x0t MSU Physics 231 Fall 2014#Clicker QuizGalileo and Newton stand on top of the tower of Pisa.

Galileo drops a stone of mass 2 kg straight down (no initial velocity).

Newton throws a 2 kg stone with an initial horizontal velocity of 3 m/s.

Which stone will hit the ground first? (ignore effects of friction)

a) The stone thrown by Galileob) The stone thrown by Newtonc) Both stones arrive at the same timed) Not enough information to say.MSU Physics 231 Fall 2014#Clicker QuizGalileo and Newton stand on top of the tower of Pisa.

Galileo drops a stone of mass 2 kg straight down (no initial velocity).

Newton throws a 2 kg stone with an initial horizontal velocity of 3 m/s.

Which stone will hit the ground first? (ignore effects of friction)

a) The stone thrown by Galileob) The stone thrown by Newtonc) Both stones arrive at the same timed) Not enough information to say.MSU Physics 231 Fall 2014#Clicker Quiz Firing Balls IA cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?a) it depends on how fast the cart is movingb) it falls behind the cartc) it falls in front of the cartd) it falls right back into the cartMSU Physics 231 Fall 2014#33Clicker Quiz Firing Balls IA small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?a) it depends on how fast the cart is movingb) it falls behind the cartc) it falls in front of the cartd) it falls right back into the cartwhen viewed from cartwhen viewed from groundIn the frame of reference of the cart, the ball only has a vertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocity, so the ball still returns into the cart.

MSU Physics 231 Fall 2014#34

MSU Physics 231 Fall 2014#Relative motionMotion is relative to a reference frame!Motion of the ball inrest-frame of cartMotion of the cartResulting motionMSU Physics 231 Fall 2014#When studying motion in 2D it is often convenient to decomposition the motion into horizontal and vertical components, both of which are both described in 1D Remember that the object can accelerate in one direction, but remain at the same speed in the other direction. Remember that after decomposition of 2D motion into horizontal and vertical components, you should investigate both components to understand the complete motion of a particle.

After decomposition into horizontal and vertical directions, treat the two directions independently but the time is common to both.MSU Physics 231 Fall 2014#Parabolic motion

MSU Physics 231 Fall 2014#v = (vx, vy)vx=v0 cosvy=v0 sinParabolic motiont=0t=4t=3t=2t=1vx=v0 cosvy=v0 sin-1gvx=v0 cosvy=v0 sin-2g=0vx=v0 cosvy=v0 sin-3gvx=v0 cosvy=v0 sin-4gVx remains constant throughout the flight! vx(t) = v0x vy (t) = v0y g t

MSU Physics 231 Fall 2014#You throw a ball in an arch (up is the positive y direction), at the highest point in the arc the y component of the acceleration is:

Impossible to tell-9.8 m/s29.8 m/s20Extra Credit QuizMSU Physics 231 Fall 2014#

t=0t=3t=2t=1t=5Where is the speedhighest?AEDCBParabolic motionMSU Physics 231 Fall 2014#

t=0t=3t=2t=1t=5Where is the speedlowest ?AEDCBParabolic motionMSU Physics 231 Fall 2014#Questionv0vfA hunter aims at a bird that is some distance away and flying very high (i.e. consider the vertical position of the hunter to be 0), but he misses.

If the bullet leaves the gun with a speed of v0 and friction by air is negligible, with what speed vf does the bullet hit the ground after completing its parabolic path?

MSU Physics 231 Fall 2014#AnswerFirst consider the horizontal direction:v0x = v0cos()Since there is no friction, there is no change in the horizontal component: vx (tf) = v0xNext the vertical direction with equations A and B:vy(t) = v0y - gt y(t) = voy t - gt2 (g = 9.8 m/s2) When the bullet hits the ground y(tf)=0:0 = voytf - gtf2 tf = 0 or tf = 2v0y/g So, vy(tf) = v0y - g(2v0y/g) = - v0yTotal speed = v0x2 + (-v0y)2 = v0!!!!The speed of the bullet has not changed, but the vertical component of the velocity has changed sign.MSU Physics 231 Fall 2014#Clicker Quiz Dropping a PackageYou drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will:a) quickly lag behind the plane while fallingb) remain vertically under the plane while fallingc) move ahead of the plane while fallingd) not fall at allMSU Physics 231 Fall 2014#45

What does the motion of the object look like according to the pilot?MSU Physics 231 Fall 2014#Clicker Quiz Dropping a PackageYou drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will:a) quickly lag behind the plane while fallingb) remain vertically under the plane while fallingc) move ahead of the plane while fallingd) not fall at allBoth the plane and the package have the same horizontal velocity at the moment of release. They will maintain this velocity in the x-direction, so they stay aligned.MSU Physics 231 Fall 2014#47Another examplex(t) = v0 cos()ty(t) = v0 sin()t gt2

(A)(B)(C)(D)(E)A football player throws a ball with initial velocity of 30 m/s at an angle of 30o degrees w.r.t. the ground.1) How far will the ball fly before hitting the ground? 2) What about an angle of 60o?3) At which angle is the distance thrown maximum?

y(t) = 0 ift( v0 sin() - gt )=0tf=0 or tf = 2 v0 sin()/gx(tf) = v0 cos() tf= 2 (v0 )2 cos() sin() / g if = 30o x = 79.5 mif = 60o x = 79.5 m !! Maximum if cos() sin() is maximum, so =45o x(=45o) = 91.7 mMSU Physics 231 Fall 2014#And another example

MSU Physics 231 Fall 2014#And another example

Vertical directiony(t) = y0 + v0 sin()t - gt211 = 37 + 10.1 sin(59)t - 9.8t24.9t2 - 8.66t - 26=0t = 3.35 (solve quadratic equation)

This time correspond to how long it takes for the ball to land on the second building. MSU Physics 231 Fall 2014#And another example

Calculate how far the ball goes in the horizontal direction.

Vertical directiony(t) = y0 + v0 sin()t - gt211 = 37 + 10.1 sin(59)t - 9.8t24.9t2 - 8.66t - 26=0t = 3.35 (solve quadratic equation)

This time correspond to how long it takes for the ball to land on the second building. Horizontal directionx(t) = x0 + v0 cos() tuse time derived from vertical directionx(3.35)=0 + 10.1 cos(59) 3.35x(3.35)=17.4 mMSU Physics 231 Fall 2014#

PHY 23152Shoot the monkeyAt the moment the hunter fires, the monkeydrops off the branch. What happens?The hunter aims his gunexactly at the monkeyAt the moment he fires, the monkeydrops off the branch. What happens?monkey gets hitbullet goes over the monkeybullet goes under the monleyno ideaMSU Physics 231 Fall 2014#PHY 23153

The vertical position of the monkey is: without g (g=0) we get the blue dashed line when x=d, y = h

With g, when x=d x=dx=0y=0y=hv0

MSU Physics 231 Fall 2014#

MSU Physics 231 Fall 2014#

MSU Physics 231 Fall 2014#Uniform circular motion with constant speedConsider a car at constant speed, constrained to a circular trackVelocity is always determined by the direction the car is facing at a given time

Velocity is constantly changing, so this implies an accelerationBut its just the direction of the velocity thats changing!

rMSU Physics 231 Fall 2014#

Uniform circular motion with constant speed vr

MSU Physics 231 Fall 2014#FrequencyEarth orbits the sun with a period of 365 days.

T=365 days(24 h/day)(3600 s/h) = 3.15107 sec

Frequency: f = 1/T = 1/(3.15107 sec) = 3.1710-8 Hz

REarth travels at constant speed throughout its orbit: v = x/t

It must traverse the circumference of the orbit: c = 2 R

Thus, the speed v = c/T = 2 R / TMSU Physics 231 Fall 2014#Clicker QuestionA hummingbird with mass 4g flies northwest at a speed of 10 m/s and at an angle of 30o relative to the horizontal. It beats its wings once every 0.014 sec. Assuming it flies at constant velocity, what is the beat frequency of its wings?

0.14 Hz7.1 Hz14 Hz71 HzNone of the abovef = 1/T

T = 0.014 sec

f = 1/0.014 sec

Frequency: f = 71 HzMSU Physics 231 Fall 2014#

Uniform circular motion with constant speed vr

MSU Physics 231 Fall 2014#

Uniform circular motion with constant velocity centripetal acceleration

Which way does the centripetal acceleration point ?MSU Physics 231 Fall 2014#

We can show that: (see book and next few slides)centripetal accelerationpoints to the center or the circleMSU Physics 231 Fall 2014#

MSU Physics 231 Fall 2014#

MSU Physics 231 Fall 2014#

Sin( /2 ) = ( s/2 ) / r Sin( /2 ) = ( v/2 ) / v MSU Physics 231 Fall 2014#

( s/2 ) / r = sin( /2 )sin( /2 ) = ( v/2 ) / v ( s/2 ) / r = ( v/2 ) / v ( s/t ) v/r = ( v/t )( s/t ) = vMSU Physics 231 Fall 2014#

Uniform circular motion with constant speed

MSU Physics 231 Fall 2014#

MSU Physics 231 Fall 2014#Clicker QuestionbeginendWhich route is shorter?RedBlackThe sameDont knowMSU Physics 231 Fall 2014#