PhysicsVECTORS AND PROJECTILE

MOTION

VECTORS

• Vectors have magnitude and direction.• Vectors are represented in diagrams as

arrows.• Vectors are represented in equations as

either bold or arrows.• Quantities that are vectors: acceleration,

velocity, displacement, force, electric field• Quantities that are not vectors: speed,

distance, time

Vectors can be added to each other.

A B

Vector A added to Vector B.

Notice that when vectors are added, they are added “tip-to-tail.”

The sum of the two vectors is called the resultant, which is drawn from the tail of the first vector to the tip of the second vectors.

Resultant, R

Now let’s add three vectors and find the resultant, so that

A + B + C = R

AB

C

R

The order in which we add these three vectors does not matter.

A + B + C = B + A + C = R

AB

C

R

The resultant vector, R, is the same size and same direction as before.

This method of addition is great for making diagrams but is not very

effective for determining the actual magnitude and direction of the resultant

vector.

We need to be able to describe the magnitude and direction of each vector mathematically and trigonometrically.

Let’s return to Vector A and Vector B.

AB

A = 8 m/s at 40 degrees north of east

B = 4.5 m/s at 45 degrees south of east

We can describe these two vectors in terms of their relative lengths (given) and their directions relative to a “compass rose”.

Next, we need to determine the components of each vector, i.e.,

the part of each vector that is along the axes.

40 degrees

A = 8 m/s

Ax = 8 cos 40°

Ay = 8 sin 40°

Now determine the components of Vector B.

[Components are vectors along the x and y axis that would add to create the vector.]

B = 4.5 m/s

Bx = 4.5 cos 45°

By = - 4.5 sin 45°

To add vector A to vector B mathematically, add their “like” components to find the x and y components of the resultant.

Rx = Ax + Bx = 8 cos 40° + 4.5 cos 45° = 9.3 m/s

Ry = Ay + By = 8 sin 40° + (-)4.5 sin 45° = 2.0 m/s

These are the components of the resultant vector:

Rx = 9.3 m/s

Ry = 5.5 m/s

Draw the resultant vector, R.

R

How can we find the magnitude of R?

Right! Use the Pythagorean Theorem.

R2 = (2.0)2 + (9.3)2 and R = 9.5 m/s

Rx = 9.3 m/s

Ry = 2.0 m/s

R

How can we find the direction of R?

Right! Use the tangent function.

Tan = 2.0/9.3 so = 12°

Now we report the final answer to our question: “What is the resultant (or

sum) of vector A and vector B?”

R = 9.5 m/s at 12 ° north of east

Note: I know it’s at that angle north of east, because both the x component and the y component are positive.

AB

R

Now you try one.

Add the following vectors to find the resultant:

10 meters at 25° north of east

20 meters at 40° south of west

30 meters at 50° west of north

1. Draw your vectors on the compass rose.

W E

N

S

1. Find the components of your vectors.

W E

N

S

3. Add the x components and the y components separately.

[Don’t forget which ones are + and which ones are -.]

Rx =

Ry =

You have found the components of the resultant.

4. Draw the components of the resultant on the compass rose and draw the resultant.

W E

N

S

Also label the angle of the resultant with one of the compass directions.

5. Use the components of the resultant to calculate angle.

tan =

Finally: State the magnitude and direction of the resultant vector.

R = ___________m

at ________ degrees ___ of ___

More to come on using vectors to describe motion in 2-d……..