From a boat in the English Channel, you slowly approach the White Cliffs of Dover. You want to know how far you are from the base of the cliff. Then suddenly you see a rock break off from the top and hit the water in 3.2 seconds. From your location in the water, the top of cliff where the rock fell makes a 30 degree angle from the surface. From this information can you determine your distance from the base of the cliff? r y θ r x The angle theta defines a vector that points from the boat to the top of the cliff where rock breaks off. That angle is given as 30 0 The x and y components of r are defined in the usual way The value of r y can be determine from the fall time of 3.2 sec We know r y and the angle. What else do need to determine r x? 1
Equality of Two Vectors A B What defines a Vector? Direction and Magnitude Two vectors are equal if they have the same magnitude /length and the same direction. if A = B and they point along parallel lines All of the vectors shown are equal. Allows a vector to be moved to a position parallel to itself Both direction and magnitude, Oh Yeah 2
Adding Vectors Vector addition is very different from adding scalar quantities. When adding vectors, their directions must be taken into account. Units must be the same Graphical Methods Use scale drawings Algebraic Methods More convenient 3
p OS = OP + PS o.. S Table I am standing on the table in 131. At point O, I move along a straight line from O to point P. It just so happens that someone is also moving the table in that same amount of time from here to there. That means that the table will have moved down and so my point P will have moved down exactly the same way and so you will see me now at point S. 4
Adding Vectors Graphically 1) Draw the first vector,, with the appropriate length and in the direction specified, with respect to a coordinate system. 2) Draw the next vector also with the appropriate length and in the direction specified 3) We can use the head to tail technique, we move to the head of 4) The sum is drawn from the tail of the first vector to the head of the last vector. 5) The order does not matter. We could instead have move to the head of 5
Parallelogram Approach 1) Draw the vectors and. 2) Bring the two tails together 3) Complete the parallelogram 4) The vector sum originates at the intersection of and,and extends across the diagonal 6
Adding more than 2 Vectors Graphically When you have many vectors, just keep repeating the process until all are included. The resultant is still drawn from the tail of the first vector to the tip of the last vector. 7
Negative of a Vector The negative of a vector is defined as the vector that, when added to the original vector, gives a resultant of zero. Represented as A A A 0 From the head to tail technique, the negative of the vector must have the same magnitude, but point in the opposite direction. A A 8
Subtracting Vectors head to tail technique Special case of vector addition: If A B, then use A B Continue with standard vector addition procedure. 9
Subtracting Vectors, Method 2 Another way to look at subtraction is to find the vector that, added to the second vector gives you the first vector. A B C As shown, the resultant vector points from the head of the second to the head of the first. 10
Algebraic Method Adding Vectors Using Components: 1. Find the components of each vector to be added. 2. Add the x and y components separately. 3. Find the resultant vector. 11
Clicker Quiz The components of vectors and are as follows: A= (Ax=1, Ay= 1) and B=(Bx=2, By=4). The components of the resultant vector C=A +B are given by A) C=(Cx= 1, Cy=5). B) C=(Cx=1, Cy=5). C) C=(Cx=0, Cy=4). D) C=(Cx=3, Cy=3). E) C=(Cx=1,Cy= 3) 12
Unit Vectors Unit vectors are dimensionless vectors of unit length and is often indicated by a hat on top of the vector symbol. Provides a convenient way of expressing an arbitrary vector in terms of its components Unit vectors x and y are define to dimensionless vectors of length 1 pointing in the positive x and y directions 13
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Vector addition and subtraction are straightforward with unit vector notation What s our vector Victor?? It provides a useful means to keep track of the x and y components What is the sum of these 3 vectors? 15
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Multiplying unit vectors by scalars: the multiplier changes the length, and the sign indicates the direction. Positive scalar does not change the direction. 17
Recall in Chapter 2, we talked about 4 different vectors in 1-D Position Displacement Velocity Acceleration These will covered again, but this time in 2-D 18
Position Vector 2-D coordinate system The position vector r points from the origin to the current location x,y 19
Displacement Vector The displacement vector r points from the original position r i to the final position r f. 20
Velocity Vector Average velocity vector: So v av is in the same direction as r. 21
Velocity Vector By taking smaller and smaller time intervals, one can calculate the instantaneous velocity vector Instantaneous velocity vector is tangent to the path and the magnitude is the speed 22