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Unformatted text preview: . PHYSICS 115 CYCLE4
WORKSHOP: MOTION IN TWO DIMENSIONS Motion in more than one dimension is a combination of the motions in each dimension. Vectors are a
mathematical tool that allow us to represent each direction of the motion as a _separate part of the motion. Vectors do not add the way that scalars (quantities without direction) do. An example would the following: A car travels east SO'miles in an hour. It then turns around and drives 10 miles west in 15 minutes. Time is a scalar, determining how long he was driving is a matter of simple addition. Displacement and
velocity are vectors, to determine his displacement and average velocity you MUST take into consideration both the distance he covered AND the direction he was moving. In one dimension we use + and — to
describe the two directions along a line. In order to describe more than one dimension we need to establish more general rules for vector addition. I. Head-to- Tail Method In the following questions ﬁnd the resultant vectors using the head-to—tail method You will need to [V :2 I -
ﬁrst determine a scale, what‘does_ 1 cm of your vector represent? Express your anSWer as the length of C”) M I the resultant vector and the angle it s ' positive-x axis, both of which you measure using a
metric ruler and a protractor. , 1. A girl walks 3 miles east then 4 miles northhrﬁu 57,, l‘ 6:; ° :
2- A girl walks 2,,miles West then 4 miles south. I Q « o \s . . 0
II. Unit Vectors and Co onents ' WW ’ H H1 ””1 3 (00 .1 In two and more dimensions it is very slow to draw head-to-tail diagrams every time we want to add
together vector quantities (for example, I. 3). For this reason we use a special mathematical "
representation called unit vectors. Unit means that the vectors are 1 unit long (i. e. 1 meter, 1 m/s, etc.) Magnitude and Direction of a
Vector Using Vector
. 0 i is a unit vector that points in the +x- components
Adding Vectors, direction. ' . .ﬁ'
- j is a unit vector that points lathe +y- They‘magnitude of a vector is its ;
direction. length It can be found using the .; Vectors are added
component by component. Pythagorean theorem If you multiply a number by a unit vector your result is a A=A i+A j
B=B:i+B:j vector quantity that points in the direction of the unit vector. The direction of a vector can be That which is multiplied by i is called the x—component of the found usmg
vector, and that which is multiplied by j is called the y—
component of the vector. C=A+B =(Ax+Bx)i+(Ay+By) j 1. The description of 1.1 above in vector notation is 3i + 4j =C. The
magnitude of this vector is 5 miles and the angle is 53. Using vector notation only (no graphs), show how to obtain the magnitude and direction. at.
Write the description of 1.2 in vector notation of a girl who walks 2 miles west then 4 miles south . Find the magnitude and direction (see the box just above this text) and compare to the answer
you obtained in I2. I. Mei/(Tw— : 34+‘t2‘ a1 iii/515% C® 5’
7W6 (3 MOLE genome-1:: 7PM $qu M .o r2041" AAIQIiIr—‘ZTD Kr cm 7'00. i’ I Practice Using Vector Comaonentsuidding velocities . Vector addition is a very powerful mathematical tool. It allows us ,as problem solvers, to think about
problems in which several things are going on at once in multiple directions. Using Vectors we can
systematically simplify the problem to one that is easier to visualize. Take the following thatwasmentioned recently by an announcer at a football game: -754“: . .. It is a very windy day. How does the quarterback know in which direction he should
throw the ball so that it makes it to the receiver? Certainly he must have to adapt to the
wine? conditions, doesn ’t he? i As physicists and engineers we know he must but in what way? Can we predict using our models of
motion and tools of vectors?
. ’52" /s I . /%r‘ [96 Let’s 100 the following problem. You have a toy car the moves with a constant velocity vhmch. It a
’1“- : A dynavclin on a treadmill, the belt of which moves at a velocitir Vb: . . '
”4/; :ILiln “0/S ( AV (NM/N 3%1f the car travels in the opposite direction to the belt, what is the magnitude and direction of the v
car’s velocity with respect to you? Although this is a one-dimensional problem, use vector
notation to represent how the velocity vectors add. 0 NW0“
MAé/eruag -‘ .5I2.m Magma») = W 5/3 / 2. Let’s actually test the model of constant velocity in two dimensions. Predict where the car will
land if it is launched across the treadmill at an angle 6. The width of the belt is w. Assume that up / -, / is the positive y-direction, and to the right is the positive x-direction. Your prediction should be '
W“ ' / 609 ,3”? expressed in terms of “may,” vbeh, w, and 9 . VLWNUA i “'7 o Check your prediction with the instructor, then test it out on using the car and the treadmill. You
will need to determine the values of vhmb, Vben, w, and 6 . ‘ ...
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- Fall '08