AP* PHYSICS B DESCRIBING MOTION: KINEMATICS IN TWO DIMENSIONS &VECTORS · 2017-10-18 · AP*...

AP* PHYSICS B

DESCRIBING MOTION:

KINEMATICS IN TWO DIMENSIONS &VECTORS

The moment of truth has arrived! To discuss objects that move in something other than a straight line weneed vectors.

VECTORS

Vectors have both magnitude AND direction while scalars have only magnitude.Velocity, displacement, force and momentum are vectors.Speed, mass, time and temperature are scalar quantities.! Vectors--arrows drawn to show direction and the length of the arrow is % to magnitude

ADDITION OF VECTORS

Graphical, tedious method--tip to tail. If the motion or force is along a straight line, simply add the two ormore lengths to get the resultant.

More often, the motion or force is not simply linear. That’s where trig. comes in. You can use the tip to tailgraphical method, BUT you’ll need a ruler and a protractor.

! Resultant--the vector sum.

1D = 1.0 km East

2D = 5.0 km North

RD = resultant and no, it’s not 6.0 km NE!!

To get the resultant, draw a picture and usetrigonometry to get BOTH the magnitude anddirection of the resultant. REMEMBER to report theangle in the “lesser” fashion. 44E N of E rather than46E E of N.

R 1 2D = (D + D ) = (100 + 25 km ) = 11.2 km2 2 ½ 2 ½

Pythagorean’s theorem is your friend!

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© 2009 by René McCormick. All rights reserved

Kinematics in Two Dimensions & Vectors 1

! To get the direction (angle):

Use trig. functions-- “Oscar Had A Heap Of Apples” is my favorite mnemonic for sin, cos, and tan (Use your own favorite!)

O = Opposite = sin H Hypotenuse

A = Adjacent = cos H Hypotenuse

O = Opposite = tan A Adjacent

! HINT: Use the data given, don’t be so cocky as to use the resultant you calculated, if you made anarithmetic error, you’re setting yourself up to make a second one!

They gave me the opposite and adjacent sides to è, the angle I wantto use in reporting my direction.

� Tan è = Opposite = 5 (km / km leaves this unitless) = 0.5 Adjacent 10

è = tan 0.5 = 27E LOOK AT YOUR PICTURE! It’s 27E N of E-1

[as opposed to 63E E of N–it is customary to report the “lesser”measure]

! All vectors have positive magnitude! If a vector is NEGATIVE, the (!) refers to its direction

SUBTRACTION OF VECTORS

!V means same magnitude, opposite direction


MULTIPLICATION OF A VECTOR BY A SCALAR V × scalar quantity = magnified magnitude, same direction as before if scalar quantity is positive, oppositedirection as before is scalar quantity is negative.

ADDING VECTORS BY COMPONENTS

We can “split” a vector into x-components and y-components and use a chart to solve for resultant vectorsquite easily. What’s a component? Consider the vector V shown in diagram (a) below:

Start by drawing a set of axes at the “head” of the vector (its origin if you prefer). The x and y-components liealong their respective axes (b) and their magnitudes are equal to:

y � V = V sin è

x � V = V cos è

You’ll come back to this method over and over between now and May! We’ll work some examples in a bit.

For now, here’s more useful trig stuff with which to wow your geeky friends [or just me!]

This Pythagorean Relationship also comes in handy from time to time: sin è + cos è = 12 2

Law of sines: A

c b

B a C

Law of cosines:

Both of these laws are handy when you have a triangle that is not a right triangle!


Example 1

A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction to the next town. Shethen drives in a direction 60.0E south of east for 47.0 km to another town. What is her displacement from thepost office?

Example 2

An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg issoutheast for 440 km; and the third leg is 53E south of west, for 550 km. What is the plane’s totaldisplacement?


PROJECTILE MOTION

Tiger Woods hitting a golf ball, a thrown or batted baseball, kicked footballs, speeding bullets, and athletesdoing the long jump or high jump are all examples of projectile motion.

OBJECTS PROJECTED PARALLEL TO THE GROUND/SURFACE

! We will often ignore air resistance (be thankful!)! We will only consider the motion after the projection (again be thankful!)! Galileo was the first to analyze projectile motion.! To keep it simple, we assume the motion begins at t = 0 and at the origin of an x,y coordinate system

o oso x = y = 0! Treat the 2 components separately, and you can apply the kinematics equations from last chapter to

each component

Let’s consider a tiny ball rolling off a table. It’s velocity vector at each instant is tangent to the path.

y! Consider v : Initially it’s zero, then gravity kicks in so now, it’s

yv = !1/2 gt 2

(if you assign “up” as positive, then y + 8 and !9)

x! Consider v : Initially it’s zero, it experiences no acceleration in

xthe x-direction, therefore its velocity remains constant and v = 0

! Gravity prevails! An object projected horizontally will reach theground in the same time as an object dropped vertically since thevertical motions are the same in each case. Dazzle your slackerfriends with this new found knowledge!

OBJECTS PROJECTED AT AN UPWARD ANGLE

yoNow you’ve gone and done it! You have introduced an initial vertical velocity component, v

! Because of the downward acceleration of

ygravity, v continually decreases (slows) untilthe object reaches its highest point on its path, at

ywhich point v = 0. Yep, you guessed it, nowthat sucker starts to accelerate downward

x! Good news! v remains constant. Make peacewith this concept. Now. Right now.

! How can this be? (It know, it seems too easy andyou are a suspicious lot by nature, huh?) The answer is: Simply because no horizontal force is acting on the projectile in the x-direction, � velocity in the x-direction remains constant!


SHALL WE JUMP OFF A CLIFF?

! Use the kinematic equations separately for eachcomponent, vertical and horizontal (see charts below)

! Simplifications:

xa = 0

ya = !g = ! 9.8 m/s2

Choose è relative to x-axis and:

xo ov = V cos è

yo ov = V sin è

Example 3

A movie stunt driver on a motorcycle speeds horizontally off a 50.0 m high cliff. How fast must themotorcycle leave the cliff-top if it is to land on level ground below, 90.0 m from the base of the cliff wherethe cameras are?


Example 4

oA football is kicked at an angle è = 37.0E with a velocity of 20.0 m/s. Calculate

a) the maximum height

b) the time of travel before the football hits the ground

c) how far away it hits the ground

d) the velocity vector at the maximum height

e) the acceleration vector at maximum height. Assume the ball leaves the foot at ground level.

Example 5

A child sits upright in a wagon which is moving to the right atconstant speed as shown in the figure. The child extends herhand and throws an apple straight upward, while the wagoncontinues to travel forward at constant speed. If air resistance isneglected, where will the apple land?


Example 6

A boy on a small hill aims his water-balloon slingshothorizontally, straight at a second boy hanging from atree branch a distance d away. At the instant the waterballoon is released, the second boy lets go and falls fromthe tree, hoping to avoid being hit. Show that he madethe wrong move!

LEVEL RANGE FORMULA:

o FOR USE WHEN y = y ONLY!

Where è is the angle made with the horizontal x-axis.

You will always get 2 answers (roots) è and 90E ! è.

Example 8

Suppose the football in Ex. 4 was a punt and left the punter’s foot at a height of 1.00 m above the ground.

o o How far did the football travel before hitting the ground? Set x = 0 and y = 0.


PROJECTILE MOTION IS PARABOLIC

That means y is a function of x an has the form y = ax ! bx , where a & b are constants for any specific2

parabolic motion. Make friends with your graphing calculator...

RELATIVE VELOCITY--The main reason we need vectors!

Example 10

A boat’s speed in still water is 1.85 m/s. If the boat is to travel directly across a river whose current has speed1.20 m/s, at what upstream angle must the boat head?

Example 11

The same boat now heads directly across the stream whose current is still 1.20 m/s.

a) What is the velocity of the boat relative to the shore?

b) If the river is 110 m wide, how long will it take to cross and how far downstream will the boat be then?


Example 12

A plane whose airspeed is 200 km/h heads due north. But a 100-km/h northeast wind (that is, coming fromthe northeast) suddenly begins to blow. What is the resulting velocity of the plane with respect to the ground?


AP* PHYSICS B DESCRIBING MOTION: KINEMATICS IN TWO DIMENSIONS &VECTORS · 2017-10-18 · AP*...

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Transcript of AP* PHYSICS B DESCRIBING MOTION: KINEMATICS IN TWO DIMENSIONS &VECTORS · 2017-10-18 · AP*...