Physics - Motion and VectorsPhysics - Motion and Vectors

CHHS - Mr. Puckett

Velocity and AccelerationVelocity and AccelerationObjectivesObjectives

Define velocity and acceleration operationally.

Relate the direction and magnitude of velocity and acceleration vectors to the motion of objects.

Create pictorial and physical models for solving motion problems.

REFERENCE FRAMES AND REFERENCE FRAMES AND DISPLACEMENTDISPLACEMENT

Frames of reference--standard for comparison. With respect to any movement of position, distance, or speed is made against a frame of reference. With respect to the Earth@ is most common.

Frame of Reference Frame of Reference

Consider the following case of the aircraft moving in relation to the earth and then the missiles moving relative to the plane and then the earth.

Various Reference FramesVarious Reference Frames

1. The Earth.

2. Cartesian Coordinate Axis : X, Y and Z – three perpendicular number lines used in all of math and geometry. Origin is zero.

3. Up (+) and Down Vertical (-)

4. Cardinal points on a compass --North, South , East, and West

VECTORS vs SCALARSVECTORS vs SCALARS

Vectors have BOTH magnitude (size) and direction. The are represented by arrows.

Vectors have both magnitude AND direction while Scalars have only magnitude.

Velocity, displacement, force and momentum are vectors.

Speed, distance, mass, time and temperature are scalar quantities.

Vectors--arrows drawn to show direction and the length of the arrow is the magnitude

Vector vs. ScalarVector vs. Scalar

When measuring motion we must distinguish between a vector quantity and a scalar quantity.

A scalar has only magnitude.

A vector has both magnitude and direction.

DisplacementDisplacement

Displacement is a change in position.

Distance vs. DisplacementDistance vs. Displacement Distance--total ground

covered--consider your route to school--total mileage put on your car/bike/feet. Scalar

Displacement--change in position of object from starting point only– “as the crow flies” from your house to school. Vector

Going in a circle yields 0 displacement and 2πr distance.

What is Velocity?What is Velocity?

Velocity is a measure of the speed and direction of the motion of an object.

As it is measuring both speed and direction, it is a vector quantity.

The velocity magnitude is given as displacement over time.

Average VelocityAverage VelocityAVERAGE VELOCITYYou can no longer use the words speed and velocity

casually. They have very specific meanings. Speed--how far an object travels in a given time; how fast

it is moving. It is a scalar quantity. average speed = distance traveled = d = meters

time elapsed t second

Average Velocity PictureAverage Velocity Picture

Constant VelocityConstant VelocityVelocity that does not change is constant

velocity.

INSTANTANEOUS VELOCITYINSTANTANEOUS VELOCITY

Instantaneous velocity--The average velocity over an infinitesimally short time interval.

The slope of the position /time graph at one place is the instantaneous velocity. This is calculus.

v = limit Δx Δt->0 Δt That limit thing just means we want Δx as close to

zero as possible without it being zero.

InstantaneousInstantaneous Velocity Velocity

Average vs. InstantaneousAverage vs. Instantaneous

Average velocity is velocity over a time interval.

v ≡ Δd/Δt = (d1-d0) /

(t1-t0).

Note that the average speed is the ratio of the total distance traveled over the total time, and is a scalar.

Instantaneous velocity is at a specific point in time. One of the problems with the average velocity is that it tells what happens over a time interval. It does NOT tell what happened DURING the interval.

v, with no bar over it, is the instantaneous velocity.

Negative VelocityNegative VelocityNote that the cars on the

bridge are going both directions. This would be negative velocity.

AccelerationAcceleration Acceleration--the rate of

change of a velocity. A change of velocity with time. If an objects velocity is changing, its accelerating. Even if its slowing down! CAREFUL of the signs.

This is also a vector quantity since it has direction. If you are changing direction, then you are accelerating.

Acceleration ExampleAcceleration Example

Calculation AccelerationCalculation Acceleration

Average Acceleration = the change of velocity divided by the change in time. The formula is : a= Δv / Δt

Example: If you are driving and start from a stop sign (v=0) and accelerate for 5 seconds and have a velocity of 25 m/s then your acceleration is (25 m/s – 0 m/s divided by (5 sec – 0 sec) for an average acceleration of 5 m/s2

Acceleration FormulaAcceleration Formula Average acceleration = a = change of velocity = Δv = meters

time elapsed Δt second2

Instantaneous AccelerationInstantaneous Acceleration

Instantaneous acceleration---The average change in velocity over an infinitesimally short time interval.

a = limit Δv Δt0 Δt NOTE THAT ACCELERATION TELLS US

HOW FAST THE VELOCITY CHANGES, WHEREAS VELOCITY TELLS US HOW FAST THE POSITION CHANGES.

Constant Acceleration Constant Acceleration Formulas for MotionFormulas for Motion

Uniformly accelerated motion--acceleration is constant and motion is in a straight line. Don’t attempt to use any of these equations unless acceleration is constant!

a = v - vo = Δv

t Δt

Area Under the Graph is the Area Under the Graph is the Integral Integral

The area under a speed/time graph is the distance traveled.

The area under an acceleration / time graph is the Velocity.

Velocity & Position EquationsVelocity & Position Equations

To solve for velocity of an object at a certain time with constant acceleration:

vf = vo + at To calculate position of an object after a time, t,

when it’s undergoing constant acceleration. Can also show vertical Y vectors:

xf = xo + vot + 1/2 at2

Velocity Without Time KnownVelocity Without Time Known

To calculate velocity, acceleration or position when time is NOT known:

v2f = vo

2 + 2a (xf - xo)

vo equals ZERO when the object begins

its acceleration from rest--this is your friend! It simplified things!

Kinematics Summary TableKinematics Summary Table

These equations can be used to calculate when acceleration is constant.

Free Fall Free Fall Free Fall is

constant acceleration toward the earth. In intro physics we ignore air drag.

Formula here is dy = ½ gt2 because it started with 0 m/s vertical velocity.

Free Fall and GravityFree Fall and Gravity

The most famous constant acceleration is that due to gravity. Memorize its value a = g = -9.80 m/s2 = -32 ft/s2.

What falls faster, a rock or a feather?– Neither, in a vacuum. Your experience is that the

feather would fall more slowly. That=s entirely due to air resistance.

Galileo--Father of Modern Science. It was he that stated at a given location on Earth and in the absence of air resistance, all objects fall at a constant acceleration, g, 9.80 m/s2.

Free Fall with gravityFree Fall with gravity

Gravity causes all objects to accelerate toward the earth at 9.8 m/s2

Objects in free fall will not accelerate forever; air drags on the object and slows the acceleration to a constant velocity called : “ Terminal Velocity”

About 120 mphFor humans

Calculating Velocity of a Calculating Velocity of a Falling ObjectFalling Object

1. We ignore the drag of air in our calculations. (Calculus-changing rates)

2. Equation: v = gt means velocity of a falling body is the acceleration of gravity times the fall time.

3. Example: If you drop a rock off a 500 m cliff: How fast is it going after 3 seconds? V = gt = (10m/s2) X 3sec = 30 m/s

Parasitic AirParasitic AirDrag Drag When astronauts

went to the moon they dropped a hammer and feather and they fell at the same rate. There was no air to slow the feather down.

Terminal Velocity & G-ForcesTerminal Velocity & G-Forces

! The speed of a falling object in air or any other fluid does NOT increase indefinitely. If the object falls far enough, it will reach a maximum velocity called the terminal velocity.

! Acceleration due to gravity is a vector (as is any acceleration) and its direction, is downward, toward the center of the Earth.

! The acceleration of rockets and fast airplanes is often expressed in g’s. Three g’s is equal to 3 x 9.8 m/s2 = 29.4 m/s2.

Ball TossBall Toss

A vertical ball toss undergoes constant acceleration but variable velocity.

Straight Up and Down Straight Up and Down KinematicsKinematics

Apex is the highest point of the trajectory above the ground where a ball stops. At that point the vertical velocity is = ZERO Acceleration is gravity.

Time to top of trajectory: T ½ = -voy /g

Total Time aloft from ground = - 2voy / g

Apex Formula = dyf = yo + voyt + ½ gt2 or

dyf = - voy2 / 2g

Vertical Motion Problem TypesVertical Motion Problem Types

1. Drop Problem: Viy = 0. dy=½gt2 and vf

= vo + gt.2. Ground to ground: Time to top (T½) =

-Voy/g, Total time aloft (Tt) = T½ x 2 . Dapex= vot + ½gt2

3. Elevated Ground to ground: Starts on elevated position up to apex Dapex=Yo + vot + ½gt2 and then a drop problem on downside.

The Cliff TossThe Cliff Toss Two examples of free fall motion are

shown in the following cliff toss. Each will vary in time aloft and final velocity.

A. The pellet is shot down at 30 B. The pellet is shot up at 30 m/s and

then falls back down with equal velocity.

C. A third classic problem is the horizontal throw starts with 0 m/s vertical velocity and drops.

ADDITION OF VECTORSADDITION OF VECTORS Graphical, -tip to tail. If the motion or force is

along a straight line, simply add the two or more lengths to get the resultant.

Graphical Non – Graphical Non – Parallel Vector Parallel Vector AdditionAddition

More often, the motion or force is not simply linear. That’s where trig. comes in. You can use the tip to tail graphical method, BUT you’ ll need a ruler and a protractor.

Trigonometry FunctionsTrigonometry Functions ! Use trig. functions-- a mnemonic for sin, cos,

and tan is SOH CAH TOA. O = Opposite = sin H Hypotenuse A = Adjacent = cos H Hypotenuse O = Opposite = tan A Adjacent

Mathematical Addition of Non-Mathematical Addition of Non-Perpendicular VectorsPerpendicular Vectors

1. Resolve initial vectors into the horizontal (Vix = Vi Cosθ) and vertical (Viy = Vi Sinθ) components. This is Vector Resolution.

2. Add the x components from the different vectors for an X total. Repeat with y.

3. Use Pythagorean to add the x and y totals: R2 = X2 + Y2 and this is the Resultant.

4. Use Tangent to find the angle: Tan θ= Y total Xtotal

5. The Resultant and angle θ are the Vector Sum.

Vector ResolutionVector Resolution

Horizontal (Vix = Vi Cosθ)

and vertical (Viy = Vi Sinθ)

These are added to get the Resultant vector.

Distance CalculatedDistance CalculatedThe formula for distance is

the (constant or average) velocity multiplied by the time you move.

D = V x t many physics books have the variable of distance as “s”

Distance is also the area under the curve graph on a velocity / time graph.

Projectile MotionProjectile MotionThe natural

motion of an object that is thrown/launched is called projectile motion.

Projectile Motion Vectors and Projectile Motion Vectors and DisplacementDisplacement

Vector ResolutionVector Resolution

Horizontal (Vix = Vi Cosθ)

and vertical (Viy = Vi Sinθ)

These are added to get the Resultant vector.

Calculate Projectile MotionCalculate Projectile Motion Range = horizontal distance traveled by the

trajectory of a projectile. We ignore air friction: = constant velocity. Range Formula: Either R = dx = vi cosθt (time aloft) from d = vt before.

Apex is the highest point of the trajectory above the ground. Acceleration is gravity. Apex Formula = Dy = yo + voyt + ½ gt2 or = - vo

2 / 2g

Time to top of trajectory: T ½ = -voy /g

Total Time aloft = - 2visin θ / g

Perpendicular Vector Perpendicular Vector IndependenceIndependence

Note in the diagram below that the initial horizontal velocity varies and it changes the range that the ball travels. But the Vertical vector remains constant throughout all shots

Constant Acceleration Graph Constant Acceleration Graph and Formulasand Formulas

Review Velocity FormulasReview Velocity Formulas 1. v = vo + at this one has initial and final

velocity, time and acceleration. 2. x = xo + vot + ½ at2 This one has initial

and final distance, velocity, time and acceleration.

3. v2 = vo2 + 2a (x-xo) This one has initial

and final velocity, acceleration and initial and final distance.

4. a = ∆v / ∆t This one has acceleration, velocity and time.

5. V = ∆d/∆t this one has velocity, distance and time

Graphing MotionGraphing Motion

One of the best ways to describe motion is with graphs.

There are 3 kinds of graphs we need to look at:– Position vs. Time graphs– Velocity vs. Time graphs– Acceleration vs. Time graphs

Position Vs. Time GraphPosition Vs. Time Graph

In this graph you are graphic the physical location vs. time for an object.

The slope of the graph is the velocity. v = Δd / Δt

Velocity Vs. Time GraphVelocity Vs. Time Graph

This graph shows the velocity of an object at any point in time.

The slope of the graph is Acceleration. a = Δv/ Δt The area under the curve is the distance

traveled.

Acceleration Vs. Time GraphAcceleration Vs. Time Graph

This graph shows the acceleration of an object at any point in time.

The area under the curve is the velocity.