1. PHY 231 1 PHYSICS 231 Lecture 4: Vectors Remco Zegers Walk-in hour: Thu. 11:30-13:30 Helproom

2. PHY 231 2 Vectors and Scalars 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

3. PHY 231 3 Question Are these two vectors the same? Are the lengths of these two vectors the same? Two vectors are equal if both their length and direction are the same!

4. PHY 231 4 Vector addition A B A+B A B B+A A+B=B+A

5. PHY 231 5 Vector subtraction A -B A-B=A+(-B) B

6. PHY 231 6 Vector operations in equations A B A+B y x (x a,y a ) (x b,y b ) (x a+b,y a+b ) Example:

7. PHY 231 7 Question begin end Which route is shorter?

8. PHY 231 8 The length of a vector and its components Y x (x a,y a ) Length of vector (use pythagorean theorem): 

9. PHY 231 9 Question A 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? B) What is the total distance he walked? 1 km 2.5 km 3 km x 2 =2.5cos(  )=1.77  =315 o Y 2 =2.5sin(  )=-1.77 A) B) 1+2.5+3=6.5 km

10. PHY 231 10 Relative motion Motion is relative to a frame! A woman in a train moving 50 m/s throws a ball straight up with a velocity of 5 m/s. A second person watches the train pass by and sees the woman through a window. What is the motion of the ball seen from the point of view from the man outside the train? Motion of the ball in rest-frame of train Motion of the train Resulting motion

11. PHY 231 11 Question  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. 1) At what angle  does the captain have to steer the boat the go straight across? A) 30 o B) 45 o C) 0 o D) -45 o 2) how long does it take for the boat to cross the river? A) 6 min B) 6.9 min C) 12 min D) 1 h 3) If it doesn’t matter at what point the boat reaches the other side, at what angle should the captain steer to cross in the fastest way? A) 30 o B) 45 o C) 0 o D) -45 o

12. PHY 231 12 Answer  Counter balance flow=5km/h Flow=5km/h Maximum v=10 km/h Hyp. Opp. 1) sin  =opposite/hypothenuse =5/10=0.5  =sin -1 0.5=30 o adjacent 2)tan  =opposite/adjacent tan30 o =0.577=5/velocity hor velocity hor =8.66 km/h time=(1 km)/(8.66 km/h)= 0.115 h=6.9 min 3) 0 o (the horizontal component of the velocity is then maximum.