Vector Example Problems 1. A man walks 10 m to the East. He then walks another 5 m to the East. What is his resultant displacement? Show with a diagram (It need not be to scale) Represent vectors with arrows: 10 m North 5 m North Resultant, R R = 15 m North
2. A man walks 40 m to the North (0o), and then turns and walks 30 m to the East (90o). What is his resultant displacement (magnitude and direction). Draw a sketch. Navigational bearing: North, 0o West, 270o East, 90o South, 180o
2. A man walks 40 m to the North (0o), and then turns and walks 30 m to the East (90o). What is his resultant displacement (magnitude and direction). Draw a sketch. 0o 30 m 270o 90o 40 m R 180o
2. A man walks 40 m to the North (0o), and then turns and walks 30 m to the East (90o). What is his resultant displacement (magnitude and direction). Draw a sketch. 30 m To find the magnitude, use the Pythagorean Theorem: 40 m R a2 + b2 = c2 ( 40 )2 + ( 30 )2 = 2 R R : magnitude of R
2. A man walks 40 m to the North (0o), and then turns and walks 30 m to the East (90o). What is his resultant displacement (magnitude and direction). Draw a sketch. 30 m To find the magnitude, use the Pythagorean Theorem: 40 m R a2 + b2 = c2 ( 40 )2 + ( 30 )2 = 2 R 1600 + 900 = 2 R 2500 = 2 R R = 2500 R = 50 m
2. A man walks 40 m to the North (0o), and then turns and walks 30 m to the East (90o). What is his resultant displacement (magnitude and direction). Draw a sketch. 0o 30 m 270o 90o To find the direction, use SOH-CAH-TOA: 40 m 50 m 180o x opp 30 tan x = = = 0.75 adj 40 x = tan -1 ( 0.75 ) = 37o
30 m Reconcile to compass: Navigational bearing = 0 + 37o = 37o 40 m R = 50 m at 37o 180o
3. A plane flying at 100.0 m/s at 90o encounters a wind blowing at 30.0 m/s at 180o. Determine the resultant velocity of the plane (magnitude and direction). Draw a sketch. 0o 100 m/s 270o 90o 30 m/s R 180o Magnitude: a2 + b2 = c2 ( 100 )2 + ( 30 )2 = 2 R 10 000 + 900 = 2 R 10 900 = 2 R R = 10 900 R = 104 m/s
3. A plane flying at 100.0 m/s at 90o encounters a wind blowing at 30.0 m/s at 180o. Determine the resultant velocity of the plane (magnitude and direction). Draw a sketch. 0o 100 m/s x 270o 90o 30 m/s 104 m/s 180o Direction: opp 30 tan x = = = 0.30 adj 100 x = tan -1 ( 0.30 ) = 17o
R = 104 m/s Reconcile to compass: 90o + 17O Reconcile to compass: 270o 90o 17O 180o Navigational bearing = 90 + 17o = 107o R = 104 m/s at 107o
4. A certain boat can travel at 20.0 km/hr in calm water. The boat is aimed straight across a river whose current is flowing at 8.0 km/hr. (a) What is the resultant velocity of the boat? 20 km/hr Magnitude: a2 + b2 = c2 8 km/hr ( 20 )2 + ( 8 )2 = 2 R R 464 = 2 R R = 464 = 22 km/hr
4. A certain boat can travel at 20.0 km/hr in calm water. The boat is aimed straight across a river whose current is flowing at 8.0 km/hr. (a) What is the resultant velocity of the boat? 20 km/hr Magnitude: a2 + b2 = c2 x 8 km/hr ( 20 )2 + ( 8 )2 = 2 R R 90o – 22o = 68o 464 = 2 R R = 464 = 22 km/hr Direction: opp 8 tan x = = = 0.40 adj 20 x = tan -1 ( 0.40 ) = 22o R = 22 km/hr at 68o to the bank
(b) If the river is 1.0 km wide, how long does it take the boat to reach the other side? 20 km/hr d = v t 8 km/hr v v R d t = 1.0 km v Use the velocity of the boat across the river 1.0 km t = 20 km/hr t = 0.050 hr
(c) How far downstream is the boat when it reaches the other side? 20 km/hr 8 km/hr R d = ? Use the velocity of the boat down the river d = v t = ( 8.0 km/hr )( 0.050 hr ) d = 0.40 km
5. Determine the resultant of the following force vectors: A = 50.0 N at 180o B = 25.0 N at 90o 270o 90o 180o R Magnitude: a2 + b2 = c2 50 N ( 50 )2 + ( 25 )2 = 2 R 25 N 2500 + 625 = 2 R 3125 = 2 R R = 3125 R = 55.9 N
5. Determine the resultant of the following force vectors: A = 50.0 N at 180o B = 25.0 N at 90o 0o 270o 90o 55.9 N Direction: 27o 180o - 27o = 153o opp 25 tan x = = = 0.50 adj 50 180o x = tan -1 ( 0.50 ) = 27o Navigational bearing = 180 - 27o = 153o R = 55.9 N at 153o