Vectors Measurements With Directions!

Any measurement with a direction is a vector. 5.6 km North 4.2m/s Forward 9.8m/s 2 Downward 5.8x10 2 N To the right Measurements without directions are known as scalars. forward

The Resultant: The result of taking into account more than one vector is known as the resultant or resulting vector. rightleft resultant PHYSICS TEXTS

If vectors are linear (along the same line): Add if they are in the same direction Subtract any vectors that are in the opposite direction

Examples A man walks 6.0m East, then 2.0m West. Two women push the same direction (E) on a box with forces of 40N and 60N. 6.0m (E) 2.0m (W) R = 6.0m (E) + 2.0m (W) = 4.0m (E) 6.0m (E) 2.0m (W) R = 4.0m (E) 60N (E) 40N (E) 60N (E)40N (E) R= 60N + 40N = 100N 

Directions: North South EastWest NE SE NW SW Up Down LeftRight

Vector Diagrams Vector Diagrams: Procedure for drawing vector diagrams: 1.Start from a point of origination. 2.Draw the first vector to scale as an arrow pointing in its direction. 3.Draw any additional vectors (to scale) from the end of the previous vector. 4.The resultant is the distance from the origin to the end of the vectors 5.Resultant direction is from the origin pointing to the end point.

Vector Diagram #1: 40N [N] 60N [E] R [NE] 1.Start from a point of origination. 2. Draw the first vector to scale as an arrow pointing in its direction. 3. Draw any additional vectors (to scale) from the end of the previous vector. 4. The resultant is the distance from the origin to the end of the vectors 5. Resultant direction is from the origin pointing to the end point. 40N [N] + 60N [E]

Vector Diagram #2: 40m [N] 60m [E] R [NE] 1.Start from a point of origination. 2. Draw the first vector to scale as an arrow pointing in its direction. 3. Draw any additional vectors from the end of the previous vector. 4. The resultant is the distance from the origin to the end of the vectors 5. Resultant direction is from the origin pointing to the end point. 40m [N] + 60m [E] + 20m [S] + 25m [W] 20m [S] 25m [W] 20 m [N] + 35 m [E] = 40 m [NE]

A person walks 4.0m East then 3.0m South 4.0m 3.0m Resultant R 2 = A 2 +B 2 or R = A 2 + B 2 R = (3.0m) 2 + (4.0m) 2 R = 9m m 2 = 25m 2 R = 5m  5.0 m [SE] Use the Pythagorean theorem (yes, from geometry!): If vectors are at a right angle (90°):

6.0N [N] 8.0N [E] 7.5kg 8.0N [E] 6.0N [N] R=? R 2 = A 2 +B 2 or R = A 2 + B 2 R = (8.0N) 2 + (6.0N) 2 R = 64N N 2 = 100N 2 R = 10N  1.0x10 1 N [NE] Calculate the resulting Force: R =?

Problem #1: 1.A man drives 12km South then 9.0km West. a)What total distance did he travel? b)What is his resulting displacement (amount and direction)? 12km [S] 9.0km [W] R = ? km [SW] a)Distance = 12km + 9.0km = 21Km b)R = A 2 + B 2 = (12km) 2 + (9.0km) 2 R = 225km 2 R = 15 km [SW] R = 144km km 2

Problem #2: 3.An evil monkey climbs 3.0m up a ladder, then 5.0m across a scaffolding. a)What total distance did the monkey travel? b)What amount is the monkey displaced? 3.0m 5.0m a) distance = 3.0m + 5.0m = 8.0m b) Displacement = R R = A 2 + B 2 = (3.0m) 2 + (5.0m) 2 5.0m 3.0m 5.8m R = 34m 2 = m = 5.8m

Problem #3: A person puts 62N of force on a 34kg box in an Eastward direction while another person puts a 42N force on the box in a Northward direction. a)What is the resultant force (amount and direction)? b)What amount is the box accelerated? 62N [E] 42N [N] R=?N [NE] a)R = A 2 + B 2 = (62N) 2 + (42N) 2 = 3844N N 2 R = 5608N 2 = 74.88N  75N [NE] b)a = F net /m = 75N/34kg = 2.205m/s 2  2.2 m/s 2 [NE]

Lab 7 – Shopping in Baltimore Compare walking distance and displacement as you travel to 5 locations around downtown. Calculate real distance and displacement vectors using map scale. 0.4 km = 400 m On the map 400 m = … 3.0 cm, so 1 cm = … 1 cm = 130 m This is your scale conversion factor Click Here to Download Map

Example Example: Draw 2 orthogonal vectors to get from Lombard & Calvert to Fayette & Gay Lombard to Fayette  (2.3 cm)  (2.3 cm)(130 m/cm) = 300 m [N] Resultant Displacement: R = √(a 2 + b 2 ) = √( ) = = 460 m [NE] Calvert to Gay  2.7 cm  (2.7 cm)(130 m/cm) = 350 m [E] Total distance = 300 m m = 650 m

Check Your Work Check Your Work: Measure the (resultant) displacement in cm and calculate % error for each trip Ex: Lombard & Calvert to Fayette & Gay Length of resultant: 3.6 cm Displacement: 3.6 x 130 = 470 m % error = 100*|(A-P)|/A = 100*(470 – 460) / 460 = 2.2 %

For each of the following trips: Trip #1: Federal Hill Park (SW corner) to corner of Pratt St. & Light St. Trip #2: Corner of Pratt St. & Light St. to Hopkins M.C. (B’way & Orleans) Trip #3: B’way & Orleans to corner of Charles St. & Chase St. Trip #4: Corner of Charles St. & Chase St. to foot of Broadway Trip #5: Foot of Broadway to water end of Lancaster 1.Measure the orthogonal distances (ie. N, S, E, or W) in cm to the nearest tenth. 2.Convert measurements to m. (use scale 1.0cm = 130m) 3.Calculate total distance travelled. 4.Calculate displacement vector. (show work, include direction) 5.When finished, check resultant by measuring the final displacement in cm, converting to m (include direction) 6.Calculate % error using orthogonal vector resultant as ACTUAL