VECTORSChapter 3: Part 1

Intro to Vectors

• Vectors indicate direction; scalars do not.• Vectors are represented by symbols• Vectors can be added graphically

Properties of Vectors• Vectors can be moved parallel to themselves in a diagram• Vectors can be added in any order• To subtract a vector, add its opposite• ****Multiplying or dividing vectors by scalars results in a

vector

Head to Tail Method• To add/subtract two vectors you MUST ALWAYS have the

vectors in a head to tail formation.• The red vector, is the sum of the two other vectors, we

call this vector the RESULTANT vector, R.

Example• Show visually how these two vectors, A & B are added.

A

B

Answer

The red line is the resultant of the two vectors, R = A+B

Mathematical Vector Addition• We can calculate MAGNITUDE of the RESULTANT of two

vectors mathematically by using the pythagorean theorem.

• We can calculate the DIRECTION of the RESULTANT of two vectors mathematically by using the tangent function.

Mathematical Example• Vector A = 3 m• Vector B = 4 m

• What is A+B?

A

B

Another Example

An archaelogist climbs the Great Pyramid in Giza, Egypt. If the pyramids height is 136 m and its width is 2.30 x 102 m, what is the magnitude and the direction of the archaelogist’s displacement while climbing from the bottom of the pyramid to the top?

You try!

While following the directions on a treasure map, a pirate walks 45 m north, then turns and walks 7.5 m east. What single straight-line displacement could the pirate have taken to reach the treasure?

Resolving Vectors into Components• If you are given the RESULTANT vector of two vectors

then we can find the components of each of these vectors by using the sine and cosine functions for right triangles

• Sinθ = opp/hyp

• Cosθ= adj/hyp

Example• Find the component velocities of a helicopter traveling 95

km/hr at an angle of 35o to the ground.

You try!• How fast must a truck travel to stay beneath an airplane

that is moving 105 km/hr at an angle of 25o with the ground?

Multiply Vectors by a Scalar• When you multiply any vector by a scalar the result is

always a vector• Ex: If vector A = 15.2m at 65o, What is the value of 3A?

• What about 4A?• 25A?• 100A?• 35A?

Adding Vectors that are not Perpendicular

• To add vectors that are not perpendicular, you must find the components of each of those vectors and add them and then find the Resultant of their added components.

Example• A hiker walks 25.5 km from her base camp at 35o south of

east. On the second day, she walks 41 km in a direction 65o north of east, at which point she discovers a forest rangers tower. Determine the magnitude and direction of her resultant displacement between the base camp and the ranger’s tower.

You try!• A football player runs directly down the field for 35 m

before turning to the right at an angle of 25o from his original direction and running an additional 15 m before getting tackled. What is the magnitude and direction of the runner’s total displacement?

PROJECTILE MOTIONChapter 3: Part 2

2-D Motion• Until now, we have only been dealing with motion in one

dimension, now we will start working in 2 dimensions

Projectile Motion• The use of components (x-direction and y-direction)

avoids vector multiplication• Components SIMPLIFY projectile motion• We neglect air resistance AND the rotation of the Earth so

therefore ---- Projectiles follow parabolic paths

• IN GENERAL, Projectile Motion is free fall with an initial horizontal velocity

How to Solve Projectiles• First step: Create a chart that looks as follows… FOR

EVERY QUESTION YOU SOLVE

Variable X Y

Vf (final velocity)

Vo(initial velocity)

a (acceleration)

X or Y (displacement)

t (time)

THINGS TO KNOW• ALWAYS START WITH THE Y-DIRECTION … BECAUSE

Y NOT? Lol (but really)• TIME IN THE X = TIME IN THE Y, EVERYTIME.

In the y-direction…• Write your variables as follows: voy, vfy, ay, Δy & t

• MOTION IN THE Y-DIRECTION IS IN FREE FALL, meaning our objects are moving at the acceleration due to gravity…….• ay = - 9.8 m/s2, ALWAYS!!!!!!!!!!!!!!!!!!!!

In the x-direction …• Write your variables as follows: vox, vfx, ax, Δx & t

• MOTION IN THE X-DIRECTION IS CONSTANT, meaning our velocity is constant the entire time so that means …..• ax = 0 m/s2, ALWAYS!!!!!!!!!!!!!!!!!!!!

One more thing …

•X-direction and y-direction can be treated and solved independently.

Projectiles launched horizontally• When projectiles are launched horizontally, the following

is true of your variables….

Variable X Y

Vf (final velocity) 0 m/s

Vo(initial velocity)

a (acceleration) 0 m/s2 -9.8 m/s2

X or Y (displacement)

t (time)

It is launched only in the horizontal, so

your initial velocity in the y is 0.

Example (pg. 101 Sample 3D)• The Royal Gorge Bridge in Colorado rises 321 m above

the Arkansas River. Suppose you kick a little rock horizontally off the bridge. The rock hits the water such that the magnitude of its horizontal displacement is 45 m. Find the speed at which the rock was kicked.

You try! (pg. 102 #1)• An autographed baseball rolls off of a 0.70 m high desk

and strikes the floor 0.25 m away from the base of the desk. How fast was it rolling?

Projectile launched at an angle• When projectiles are launched at an angle you now have

components of your initial velocity

GROUND or SURFACE

θ

V o

Vox

Voy

Example 2 (pg. 104 #3)• A baseball is thrown at an angle of 25o relative to the

ground at a speed of 23.0 m/s. If the ball was caught 42.0 m from the thrower, how long was it in the air? How high was the tallest spot in the ball’s path?

You try! (pg 104 #2)• A golfer can hit a golf ball a horizontal distance of over

300 m on a good drive. What maximum height will a 301.5 m drive reach if it is launched at an angle of 25o to the ground? (Hint: At the top of its flight, the ball’s vertical velocity component will be zero)