TIP-TO-TAIL METHOD We will be using the graphical tip-to-tail method of finding the resultant R of several vectors.

To use this method you begin at any convenient point and drawing (to scale in the proper directions) each vector arrow in turn. The tail of each arrow is positioned at the head of the preceding one.

The resultant is represented by an arrow with its tail end at the starting point and its head at the tip of the last vector added.

Can we add two vectors mathematically?

Yes, we can use the rules of the number line and the rules of trigonometry…but FIRST…we will do it graphically.

Let’s look at some simple examples…

PSE pg. 16 Example 1: Every March, the swallows return to San Juan Capistrano, California after their winter in the south. If the swallows fly due north, and cover 200 km on the first day 300 km on the second day and 250 km on the third day, find their total displacement for the three-day trip.

These vectors are all in the same direction…

200 km

300 km

250 km

So they add…

N

200 km + 300 km + 250 km =

750 km N

EXAMPLE 2: The Smock family begins a vacation trip by driving 700 km west. Then the family drives 600 km south, 300 km east and 400 km north. Where will the Smocks end up in relation to their starting point? Solve graphically.

1 cm = 100 km 700 km

600 km

300 km

400 km

450 km 27º S of W

N

EW

S

θ

Use a ruler / protractor!

TIP-TO-TAIL METHOD We have been using the graphical tip-to-tail method of finding the resultant R of several vectors.

To use this method you begin at any convenient point and drawing (to scale in the proper directions) each vector arrow in turn. The tail of each arrow is positioned at the head of the preceding one.

The resultant is represented by an arrow with its tail end at the starting point and its head at the tip of the last vector added.

Can we add two vectors mathematically?

Yes, We can use the rules of the number line and the rules of trigonometry

Let’s look at some simple examples…

PSE pg. 16 Example 1: Every March, the swallows return to San Juan Capistrano, California after their winter in the south. If the swallows fly due north and cover 200 km on the first day 300 km on the second day and 250 km on the third day, find their total displacement for the three-day trip.

These vectors are all in the same direction…

200 km

300 km

250 km

So they add…

N

200 km + 300 km + 250 km =

750 km N

Example 2: If St. Louis Cardinals homerun king, Mark McGwire, hit a baseball due west with a speed of50 m/s and the ball encounters a wind that blew it north at 5 m/s, what is the resultant velocity of the baseball?

5 m/s

50 m/s

Let’s sketch it…R

R = √ (50 m/s)2 + (5 m/s)2 = 50.25 m/s

Tan θ = 5 m/s 50 m/s

Θ = 5.71º N of W

EXAMPLE 3: The Smock family begins a vacation trip by driving 700 km west. Then the family drives 600 km south, 300 km east and 400 north. Where will the Smocks end up in relation to their starting point? Solve mathematically.

Can we simplify this problem?N

W

S

E

+ y

- y

+ x- xWe can add all the x’s

Rx = -700 km + 300 km = - 400 km

We can add all the y’s

Ry = - 600 km + 400 km = - 200 km

Rx = - 400 kmRy = - 200 km

So we are left with only two vectors…

And they look like… 400 km

200 km

R= √ (400 km)2 + (200 km)2 R = 447.21 km

θ

Tan θ = 200 km 400 km

Θ = 26.57º S of W

Be prepared for treasure hunting next class!!