Non-linear Motion

Non-linear Motion

Motion in 2DChapter 3

Linear and Non-linear Motion

• Linear motion is motion along a straight line. It can be described with one dimension.

• Non-linear motion is motion along a curved path. It is best described with two dimensions.

x direction

x direction

-y direction

x direction+y direction

Horizontal Projectile – example of non-linear motion

• A projectile is an object upon which the only force acting is gravity

Types of Projectiles

• There are several different types of projectiles. We will only study horizontal projectiles and objects in free fall.

Horizontal ProjectileFree Fall

Examples of Horizontal Projectiles

• Package dropped by an airplane that is flying with a constant velocity in the horizontal direction http://regentsprep.org/Regents/physics/phys06/apackdrop/default.htm

• Object thrown/driven/rolling off a cliff or table http://regentsprep.org/Regents/physics/phys06/apackdrop/horizpro.htm

Challenge Question

• Which of the following will hit the ground first, a ball dropped or a ball thrown horizontally?

Motion in 2D• A projectiles motion is a combination horizontal (motion in

the x direction) and the vertical motion (motion in the y direction)

• Horizontal and vertical motion can be considered separately• http://www.physicsclassroom.com/mmedia/vectors/

mzng.html- horizontal motion only

Vectors and Scalars• In studying motion and forces in 2D it is helpful separate quantities into

those that depend on direction and those that do not• Vectors are quantities which are fully described by both a magnitude and

a direction.• Scalars are quantities which are fully described by a magnitude alone.Quantity Vector Scalar Comments

Distance X Refers to "how much ground an object has covered" during its motion.

Displacement X refers to "how far out of place an object is"

Speed X how fast something is moving

Velocity X how fast and in what direction

Acceleration X

Temperature X

Time X

Force X

Vectors• Physical quantities that have both magnitude and direction are

conveniently represented by vectors.• Vectors are represented graphically by arrows indicating their direction

and having lengths proportional to their magnitudes (size).• All vectors have a tail and head.• The resultant is the vector sum of two or more vectors.

5 m East

10 m East

10 m West

5 is the magnitude of this vector, East is the direction

These vector is twice as long because their magnitude is 10, instead of 5

Equivalent sets of vectors?• Examine the following sets of vectors and

decide if they are equivalent or not. Justify your answer in each case.

5 m East10 m East

10 m East

10 m West

5 m East

5 m South

5 m East

5 m East

a)

b)

c) d)

Vector Addition• If vectors are in the same direction, simply add the numbers to

get the resultant• If vectors are in the opposite direction, subtract to get resultant• If vectors are at a right angle, use Pythagorean theorem

Adding Velocity Vectors1) Find the resultant velocity of an airplane flying at

100 km/hr with a 25 km/hr tail wind. 100 km/hr + 25 km/hr = 125 km/hr

2) Find the resultant velocity of an airplane flying at 100 km/hr with a 25 km/hr head wind. 100 km/hr - 25 km/hr = 75 km/hr

Resultant

25 km/hr 100 km/hr

75 km/hr

Adding Velocity Vectors• The resulting flight path is a result of both velocity vectors

and can be found using a2 + b2 = c2 or a scale diagram. • 1002 + 252 = c2 c = 103.1 km/hr• http://www.physicsclassroom.com/mmedia/vectors/

plane.html RESULTANT

25 km/hr crosswind 100 km/hr direction

Vectors Question

• Which angle of attack will bring a kayaker to the other bank at a point directly opposite her entry point?

River Crossing• When finding the path of a boat that is

crossing a river that is flowing, the velocity of the boat and the velocity of the river are added to find the resultant velocity.

• http://www.physicsclassroom.com/mmedia/vectors/rb.html

Vectors and Projectile motion - Velocity• The velocity in the y-direction increases as the ball falls because the force of

gravity pulls the ball downward. It is calculated using vy = gt.

• The velocity in the x-direction does not change because gravity only acts in the up/down direction. The velocity in this direction will remain at its starting value for the entire path.

Note: Horizontal velocity vector (vx)remains the same length throughout the path but the vertical velocity vector (vy) increases in length because the vertical velocity is increases as the ball falls

Vectors and Projectile Motion - Displacement• To calculate the vertical distance traveled we use the free fall equation for

distance, d = ½ gt2 or y = ½ gt2

• To calculate the horizontal distance traveled we use x = vxt, the equation used to calculate distance when there is no acceleration

- horizontal distance traveled (x)

- Vertical distance traveled (y)

Before the ball starts to fall, at t = 0 s, it is traveling horizontally at 25 m/s

g

yt

2

To find time when vertical distance is given

Vector Questions1) A motor boat traveling 4 m/s, East encounters a

current traveling 3.0 m/s, East. What is the resultant velocity of the motor boat?

2) A motor boat traveling 4 m/s, West encounters a current traveling 3.0 m/s, East. What is the resultant velocity of the motor boat?

3) A motor boat traveling 4 m/s, East encounters a current traveling 3.0 m/s, North. What is the resultant velocity of the motor boat?

Note: Remember all answers need magnitude (a number) and direction (N, E, W, NE, SW etc.)

Vector Answers

1) 4 m/s + 3 m/s = 7 m/s East2) 4 m/s – 3 m/s = 1 m/s West3) 42 + 32 = c2 c = 5 m/s NE

Projectile Questions1) A ball rolls off a table, what factors will influence how far the

ball lands from the edge of the table?2) A person standing on the top of a building shoots a bullet

horizontally from a gun and at the same instant drops a bullet. Which one hits the ground first? Explain.

3) Does gravity act in the x (horizontal) or y (vertical) direction as a projectile moves through the air?

4) It is the vertical (y) or horizontal (x) velocity of a projectile that remains constant (does not change)?

5) A boy on the top of a 20 m tower throws a ball horizontally at 30 m/s. a) How long is the ball in the air? b) How far does the ball travel horizontally before it hits the ground?

Projectile Answers

1) The height of the table and the speed of the ball just before it leaves the table

2) They hit at the same time because they are both falling at the same rate, the horizontal velocity of the bullet does not effect the rate at which it falls.

3) y direction (up/down direction)4) Horizontal (x) velocity5) a) y = ½ gt2 y = 20 m g = 10 m/s2 t = ? = 2s

b) x = vt = (30 m/s)(2s) = 60 mg

yt

2

Satellite Motion

• How fast would the cannon ball have to travel to go around the earth in a circle?

Sources

• Conceptual Physics by Paul Hewitt• http://www.physicsclassroom.com• portal.chaminade-stl.com/LinkClick.aspx?

link=Projectile+Motion+-+Chapter+3.ppt&tabid=4749&mid=7955