Chapter 3Chapter 3

Nonlinear MotionNonlinear Motion

• Scalar quantity ----• ------ a quantity that has

magnitude but not direction

• Vector quantity -• ------ a quantity that has both magnitude and direction

• Vector - • -------an arrow drawn to scale used

to represent a vector quantity

These represent equivalent vectors:

• Vector quantity - a quantity that has both magnitude and direction

• Vector - an arrow drawn to scale used to represent a vector quantity

• Scalar quantity - a quantity that has magnitude but not direction

ExamplesExamples

• Speed………..• Velocity……....• Acceleration..• Time………….• Distance……..• Force…………

scalarvectorvectorscalarscalarvector

Addition of VectorsAddition of Vectors• The sum of two or more vectors is called their

resultant.

• To find the resultant of two vectors that are at angles to each other, we use the tip-to-tail method.

Projectile MotionProjectile Motion• A projectile is any object that is projected by some means and continues in

motion by it own inertia.

• The velocity of a projectile has a horizontal and vertical component. Each component acts independently of the other.

• For the vertical motion the acceleration is 9.8m/s2 downward.

• For the horizontal motion there is no acceleration.

• Projectile Drawing.

Projectile MotionProjectile Motion

• The shape of a projectiles path is a parabola.

• The same range is obtained from two different projection angles that add up to 90°.

• Maximum range for a projectile is achieved with a projection angle of 45°.

• In the presence of air resistance, the trajectory of a high-speed projectile falls short of a parabolic path.

• A projectile fired horizontally will hit the ground at same time as an object dropped from rest if they are released at the same height.

• Demo: Ball projector and dropper

Example QuestionsExample Questions• You are driving along in an open car and throw

a ball straight up into the air. Neglect air resistance.

(a) Where does the ball land relative to the car?Answer: In the car.

*

Example: Projectile Motion• An object may move in both the x and y directions

simultaneously (i.e. in two dimensions)• The form of two dimensional motion we will deal with is called

projectile motion

• We may: » ignore air friction

» ignore the rotation of the earth

• With these assumptions, an object in projectile motion will follow a parabolic path

Notes on Projectile Motion:

• once released, only gravity pulls on the object, just like in up-and-down motion

• since gravity pulls on the object downwards:

vertical acceleration downwards NO acceleration in horizontal direction

Projectile Motion

Rules of Projectile Motion

• Introduce coordinate frame: y is up• The x- and y-components of motion can be treated

independently• Velocities (incl. initial velocity) can be broken down into

its x- and y-components• The x-direction is uniform motion

ax = 0• The y-direction is free fall

|ay|= g

Some Details About the Rules

• x-direction – ax = 0– – x = vxot

• This is the only operative equation in the x-direction since there is uniform velocity in that direction

constantvcosvv xooxo

More Details About the Rules

• y-direction– – take the positive direction as upward– then: free fall problem

• only then: ay = -g (in general, |ay|= g)

– uniformly accelerated motion, so the motion equations all hold

ooyo sinvv

Velocity of the Projectile

• The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point

x

y12y

2x v

vtanandvvv

(b) While the ball is still in the air you step on the accelerator. Where does the ball land relative to the car?

Answer: Behind the car.

(c) What if you stepped on the brake instead?Answer: In front of the car.

• Demo: Cart and ball launcher

Example QuestionsExample Questions• You drop a ball from the window of a school bus

moving a 10 miles/hour. Neglect air resistance.

(a) Where does the ball land relative to your hand?Answer: Directly below your hand.

*

(b) What is the shape of the path made by the ball seen by someone outside the bus?

Answer: A parabola.

Fast-Moving Projectiles SatellitesFast-Moving Projectiles Satellites

• An earth satellite is simply a projectile that falls around the earth rather that into it.

• Satellites are not free from gravity.

• The speed of the satellite must be great enough to ensure that its falling distance matches the earth's curvature.

• It takes 1 second for an object in free fall to fall 4.9 meters.

• Earth's surface 'drops' a vertical distance of 4.9 meters for every 8000 meters along the Earth's surface.

• This means that a satellite near the Earth’s surface must travel at 8000 meters/second!

• …or 18,000 miles per hour.

• For example: The space shuttle orbits the Earth once every 90 minutes.

Circular MotionCircular Motion• Linear speed - the distance moved per unit

time. Also called simply speed.

• Rotational speed - the number of rotations or revolutions per unit time.

• Rotational speed is often measured in revolutions per minute (RPM).

• The linear speed is directly proportional to both rotational speed and radial distance.

v = r

Example QuestionExample Question• Two ladybugs are sitting on a phonograph

record that rotates at 33 1/3 RPM.

(a) Which ladybug has a great linear speed?Answer: The one on the outside edge.

(b) Which ladybug has a great rotational speed?Answer: Both have the same

rotational speed.

*

You sit on a rotating platform halfway between the rotating axis and the outer edge.

You have a rotational speed of 20 RPM and a tangential speed of 2 m/s

What will be the rotational speed of your friend who sit at the outer edge?

Answer: 4 m/s

What will be his rotational speed?

Answer: 20 RPM

• See this question on page 50.

*

End of Chapter 3