Vectors. However, cartesian coordinates are really nothing more than a way to pinpoint an object s position in space

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1 Vectors Definition of Scalars and Vectors - A quantity that requires both magnitude and direction for a complete description is called a vector quantity ex) force, velocity, displacement, position vector, unit vector - A quantity that is described only by magnitude is called a scalar quantity ex) mass, volume, time, temperature, numbers - A symbol for vector is described using an arrow ex) 3.1 Coordinate Systems The two coordinate systems used in physics are cartesian coordinates and polar coordinates i) Cartesian coordinates are often used to define a point in space. To define this point by starting at the origin and then moving x units horizontally followed by y units vertically However, cartesian coordinates are really nothing more than a way to pinpoint an object s position in space ii) Polar coordinates can also be used to define a point in space Another way to define this point, go straight out of the origin until we hit the point and then determine the angle this line makes with the positive x-axis. We could then use the distance of the point from the origin and the amount we needed to rotate from the positive x-axis as the coordinates of the point. Polar coordinate is useful when describing circular motion Cartesian coordinates => (x,y) Polar coordinates => (r,θ) - The relationship between cartesian and polar coordinates can be expressed using trigonometric functions => x = rcosθ, y = rsinθ r = x² + y² θ = tan 1 ( y x ) *~ the positive angle θ is measured counterclockwise from the positive x-axis~* Review Example3.1(p.60). The Cartesian coordinates of a point in the xy plane are (x,y)=(-3.50, -2.50)m. Find the polar coordinates of this point 1

2 3.3 Some Properties of Vectors i) Equality of two vectors Two vectors A and B may be defined to be equal if they have the same magnitude and point in the same direction(see fig.3.5 in pg.62 in your textbook) ii) Adding Vectors - Adding two or more vector quantities are different from adding scalar quantities. - When adding scalar quantities, we can simply add the magnitude ex) = 6, 2kg + 4kg = 6kg, 60ºF + 40ºF = 100 ºF => Scalar addition - When adding vector quantities, we must not only add the magnitude, but also take into account the direction - If you add two lengths without considering the direction, we can just simply add the magnitude of the length - If you add two lengths considering the direction, the resultant length may not simply be adding the magnitude of the length ex) i) Two equal forces(vectors) are acting on the same direction, ii) opposite direction ex) Two forces(vectors) are at i) right angles(90 ), ii) less than 90 Adding two vectors can be described using geometric methods, called triangle method of addition(see fig.3.6 in pg.62 in your textbook) The resultant vector of two vectors A and B is written as R. => R = A + B =>R is the vector drawn from the tail of the first vector to the tip of the last vector Vector addition obeys the commutative law of addition => A + B = B + A (see pg.62) Vector addition obeys the associative law of addition => A + ( B + C ) = ( A + B ) + C (see pg.63) Review Quick Quiz3.3 (pg.64) If vector B is added to vector A, which two of the following choices must be true for the resultant vector to be zero? (a) A and B are parallel and in the same direction. (b) A and B are parallel and in the opposite direction (c) A and B have the same magnitude. (d) A and B are perpendicular 2

3 *Components of a Vector and Unit Vectors Adding vectors are more accurate using components than using geometric method. Ay A A = A x + A y, where Ax = Acosθ, Ay = Asinθ and A= Ax² + Ay² θ Ax The angle θ =tan -1 A y A x The signs of the components A x and A y depend on the angle θ. => if θ =120, then A x is negative and A y is positive. If θ =225, then both Ax and Ay are negative. Displacement Vectors (or Position vectors) and Unit Vectors A vector A can be written as A = Axi + Ayj and the point lying in the xy plane can be written as r = xi +yj ex) if x=3, y=4, then r=3i + 4j and the magnitude r can be found by r = rx² + ry² = 3² + 4² = 5 Vector quantities often are expressed in terms of unit vectors. An unit vector is a dimensionless vector having a magnitude of exactly 1. Unit vectors are used to specify a given direction and have no other physical significance. They are used solely as a convenience in describing a direction in space. The symbols i,j and k will be used to represent unit vectors pointing in the positive x,y and z direction respectively. The magnitude of each unit vector equals i = j = k =1 ex) A vector A that lies on the x-axis with a magnitude can be expresses as A=3i and drawn as below. 3 i 1 3

4 Example 3.3 The Sum of two Vectors => see p.68 for answers Find the sum of two displacement vectors A and B lying in the xy plane and given by A = (2.0i + 2.0j)m and B = (2.0i 4.0j)m Also find the magnitude and direction of the resultant vector **Expressing Velocity and Acceleration using Unit Vectors Since velocity is a vector quantity, we can express the velocity using unit vectors. vyj θ =40 v vxi - The velocity can be expressed as v= v x i + v y j for 2 dimensional motion - v x and v y are the components of velocity and are scalar quantities - v x =vcos40 and v y=vsin40 where v is the actual speed of the object. - If the speed is 4m/s, then the velocity can be expressed as v= 4cos40 i + 4sin40 j = 3.06 i j (m/s) - If the velocity was first given as v= 3.06 i j (m/s), then speed of the velocity can be found by v= 3.06² ² 4m/s - The advantage of using unit vectors for the velocity is that we can see the components and find the actual speed with a single expression Q1) If an object is in free fall, the gravitational acceleration is a g = -9.8m/s 2 (downwards). To express this in unit vectors, it is a) a = (-4.8i - 4.8j) m/s 2 b) a = (4.8i + 4.8j) m/s 2 c) a = -9.8j m/s 2 d) a = -9.8i m/s 2 Q2) Which of the following velocity is moving only horizontally? a) (-12.0j)m/s b) (12i -15.3j)m/s c) (12.0i)m/s d) (25.2i -15.3j)m/s Q3) Which of the following velocity is rising vertically? a) (+12.0j)m/s b) (-12j)m/s c) (12.0i)m/s d) (25.2i -15.3j)m/s 4

5 v = v0 + at x=x0 + v0t + (1/2)at 2 v 2 = v a(x x0) Q4) A sling-shot shoots a rock at a height of 12 m(or r = -12j m ) with an initial horizontal speed of 20m/s. (or initial velocity of v =20i m/s. The expression of the velocity indirectly shows that the speed is horizontal only!) vyo=0 vy 12meters vy vy vy = gt = ( )m/s i) How long was the rock in the air before it hit the ground? (remember that if an object is shot horizontally then the time it takes to hit the ground is equal to just dropping at that same height) a) 1.56s b) 2.76s c) 4.38s d) 4.95s ii) Find the horizontal displacement of the rock. (remember that since there is no horizontal force acting on the rock, the horizontal speed v x is constant throughout the motion) a) r =16.4i(m) b) r =31.3i(m) c) r =54.4i(m) d) r =78.2i(m) iii) What is the velocity of the rock the right before it hits the ground? a) (15.3i -20.0j)m/s b) (20.0i -15.3j)m/s c) (20.0i +15.3j)m/s d) (25.2i)m/s 5

6 v = v0 + at x=x0 + v0t + (1/2)at 2 v 2 = v a(x x0) iv) What is the speed of the rock the right before it hits the ground? (Remember that the actual speed v can be solved by v = v 2 x + v2 y ) a) 11.3m/s b) 15.3m/s c) 18.3m/s d) 25.2m/s v) Find the speed and velocity(in unit vector notation) of the rock at t=0.2s, t=0.6s and t=1s 6