Tierheim Associates Dream for the stars and sail the winds of change! Tierheim Associates Linda Lanham Summitt To rejoice in the wonders all around us and to learn the voices and colors on the wind will give us strength to touch the sky.
Linear Motion • Linda L Summitt • Tierheim Associates
One Dimensional Kinematics • The bicycle is a tremendously efficient means of transportation. In fact cycling is more efficient than any other method of travel--including walking! The one billion bicycles in the world are a testament to its effectiveness. The engine for this efficient mode of transport is the human body. Because bodies are fueled by food, diet plays an important role in how the body performs. Different muscle groups and types provide the power. Genetic inheritance, intensive training, and a competitive drive help top athletes push the boundaries of endurance and speed on the bicycle.How Far Do You Want To Go? It takes less energy to bicycle one mile than it takes to walk a mile. In fact, a bicycle can be up to 5 times more efficient than walking. If we compare the amount of calories burned in bicycling to the number of calories an automobile burns, the difference is astounding. One hundred calories can power a cyclist for three miles, but it would only power a car 280 feet (85 meters)! • A comparison of the energy cost of various forms of transportation shows that the bicycle is most energy-efficient. How long should it take the average cyclist to complete the 2178 mile Tour de France?
Objectives • Define motion, speed, velocity, and acceleration • Distinguish between uniform and variable acceleration • Apply motion equations to solve problems • Use the components of motion to plot graphs • Use graphs to determine motion
AP Goals • Calculate velocity or acceleration from distance time or acceleration time graphs • Identify graphs for uniform & non-uniform motion • Distinguish between average & instantaneous velocity and acceleration • Select equations to solve for unknown motion quantities • Solve kinematic problems • Given a graph solve for kinematic quantities • Identify independent relationship of horizontal & vertical components • Find magnitude and direction of vectors and resultants • Use vector analysis to solve projectile problems • Understand the motion of a projectile
Introduction • Kinematics is the quantitative analysis of motion (it describes motion of objects) • Dynamics is the study of relationships and causes of motion
Kinematics • Kinematics is the branch of physics that concerns the study of motion without regard to cause. • Motion along a straight line is known as linear motion.
Frame of Reference • Frame of reference is determined by the choice of axis defined by the starting point.
Position, Distance and Displacement • Position is a location that can be described by coordinates (x,y) • Distance is measurement between two positions • Displacement is a change in position of an object. It is independent of the path taken. Displacement is given the designation s, Dx Dy or Dz distance displacement
Dimensional Displacement • If a rabbit runs across the yard and his x coordinates vary according to the equation x = -0.31t2 +7.2t +28 and his y coordinates vary according to the equation y= 0.22t2 –9.1t +30 find the position of the rabbit form the origin at time 0s, 5s, 10s, 15s, 20s and 25s x y T=0 28 30 T=5 56.25 -10 T=10 69 -39 T=15 66.25 -57 T=20 48 -64 T=25 14.25 -60
Displacement Dx = xf-xi • Displacement is a vector quantity requiring direction and magnitude (3 mi at 300 NE) • A motorist travels north for 35 minutes at 85 km/hr, stops for 15 minutes, then continues north traveling 130 km in 2 hrs. What is the total displacement?
Dimensional Displacement • This will give the position or coordinates of the displaced object. z r x y
Position • Position is a result of the vector components of the 3D coordinates from a reference point”r0” to a point rf Dr = rf - r0 • Dr= (xf-x0)i+ (yf –y0)j + (zf –z0)k
Magnitude and direction of displacement • After finding the coordinates of the displaced object Pythagoreans theorem and trig can be used to find the magnitude and direction of displacement
Speed • Speed is a scalar quantity describing distance traveled in a specified time. • Average Speed = Total distance / total time • Scalar quantities have magnitude only.
Average Velocity • Velocity is a vector quantity. It describe displacement with relation to time. • Vector quantities have magnitude and direction. • V = s/t = Dx/Dt = vf +vi / 2 • Average Velocity need not equal average speed.
A car makes 2.5 trips around a track 50 meters in diameter in 30 minutes. • What is the average speed of the car? • What is the average velocity of the car? C=pd: 3.14(50) so 157m x2.5 / 1800s= 0.218m/s V=Dd/Dt: 50m/1800s= .028m/s
Instantaneous Velocity • The velocity at any given point of time is known as instantaneous velocity. • It is equal to the slope of the line at the point on a straight line of a position time graph or the slope of the tangent to that point on a curved line on a position time graph. • Vinst = lim t->0Dx/Dt = dx/dt or the derivative of x with respect to t.
Example Instantaneous Velocity • An object is moving along the horizontal axis according to the equation x = -4t +2t2 . Determine the instantaneous velocity of the object at 1.5 s and 2.5 s.
Instantaneous Velocity • If x varies according to the equation x=-4t+2t2 6 4 2 0 -2 Slope dx = vdt 0 1 2 3 4
Graphing Velocity • The slope of a line on a position-time graph is the average velocity. • The slope of a tangent is the instantaneous velocity. • The area under the line on a velocity- time graph gives the displacement.
Graphs Velocity/time Position/time Instantaneousvelocity at A Displacement is area under graph lw=A vt=d 10 8 6 4 2 0 -2 -4 -6 Zero velocity 8x2=16m 4x4=16m A -2x1 =-2m 0 1 2 3 4 5 6 7 8 9 Average velocity to A 16m+16m-2m=30m
Answer Now 0 of 5 Determine the average velocity between 0 to 6 seconds 20 10 0 -10 • 10 m/s • 20 m/s • 0.1 m/s • -10 m/s 1 2 3 4 5 6 Velocity/time graph
Answer Now 0 of 5 Determine the average velocity between 1 and 5 seconds 20 10 0 -10 • 10 m/s • - 10 m/s • 20 m/s • -20 m/s 1 2 3 4 5 6 Velocity/time graph
Answer Now 0 of 5 Determine the distance traveled from 0 to 6 seconds • 55 m • 45 m • 60 m • 110 m 20 10 0 -10 1 2 3 4 5 6 Velocity/time graph
Average velocity • Average velocity of a particle moving through space is just the displacement divided by the change in time. • Instantaneous velocity is the derivative of the instantaneous velocity or the derivative of the velocity at a given time.
Finding velocity example • From the rabbit problem discussed earlier • If a rabbit runs across the yard and his x coordinates vary according to the equation x = -0.31t2 +7.2t +28 and his y coordinates vary according to the equation y= 0.22t2 –9.1t +30 find the position of the rabbit form the origin at time 0s, 5s, 10s, 15s, 20s and 25s • Find the derivative of the position to find the velocity coordinates. Plug time in to find the velocity coordinates.
Finding velocity cont’ Use Pythagoreans and trig to find the magnitude and direction of the instantaneous velocity at a given time.
Practice • If you are on a Ferris wheel that has a radius of 10 m and you ride to the top: • What distance have you traveled? • 1/2(2pr) • What is your displacement? • d • Physics B Lesson 01: Motion in One Dimension : Monterey Institute for Technology and Education : Free Download & Streaming : Internet Archive
Vectors Math • Vectors can be added or subtracted graphically or by breaking each vector into components and adding or subtracting components and then using Pythagoreans theorem. • If you need a more detailed review over Vectors go to the AP Physics Process and Measurement Presentation
Problem 1 • A person can row a boat at the rate of 8.0 km/hr in still water. The person heads the boat directly across a stream that flows at the rate of 6.0 km/hr. Find the resultant velocity. 8.0 km/hr 6.0 km/hr
How long? • If the stream is 120 m wide how long will it take the person to get across the stream?
Problem2 • A car traveling at 27 m/s passes a 920 m long train traveling a 18.3 m/s. • How long does it take the car to pass the train?
Solution 2 • Vc = 27 m/s lt = 920 m • Vt = 18.3 m/s • V=s/t • Sc= 27m/s t St= 18.3m/s t • Sc - 920m = St • Sc - 920m = 18.3m/s t • 27m/s t - 920m = 18.3m/s t • t = 106s
Uniform Linear Acceleration • When acceleration is constant and in a straight line we call it uniform linear acceleration. • Rearranging the equation for average velocity we get vf = v0 +at. • In this case average and instantaneous velocity are equal.
Acceleration • Acceleration is a change in velocity over a specified time period. • aavg = vf-vo / t. • A negative sign indicates acceleration in the negative direction or a deceleration. • If v and a are the same the object is accelerating if they are different it is decelerating. • The slope of a velocity time graph gives the average acceleration.
Signs and Acceleration • If the acceleration is positive direction and speeding up or negative and slowing down then the acceleration is positive. • If the acceleration is in the positive direction and slowing down or in the negative direction and speeding up then acceleration is negative.
Derived Equations • By substitution V = s/t and V = vf +vi / 2 if s = vt then s = 1/2 (vf +vi ) t and t = vf-vo / a so 2as = (vf +vi )(vf-vo ) or vf2 = vi2 + 2as • By substitution of vf = v0 + at into the equation s = 1/2 (vf +vi ) t we get s = 1/2 ( v0 + at) + 1/2 v0 t which simplifies into s = v0 t + 1/2 at2
Motion Diagrams • Motion diagrams are diagrams of sequential pictures taken with stroboscopic cameras. These take pictures at regular time intervals so that changes in velocity can be calculated.
Free Fall • Aristotle (384-322 BC) Heavy Objects Fall Faster! ??? • Galileo Galilee (1564-1642) Diluting Gravity experiment using an increasing angle of incline shows that acceleration is constant. DO IT YOUR SELF AND SEE WHO WAS RIGHT!
Free Fall • When an object falls with acceleration due to gravitational force and experiences no other significant forces this is known as free-fall • Earth’s gravitational force causes free-fall at the rate of 9.8 m/s2 • DIRECTION must be defined at the beginning of each problem and by convention we assign up as positive; therefore acceleration due to gravity is -9.8 m/s2 and is designated g
Motion, Practical Applications & Calculus • When a tractor trailer is driven at a speed of x miles per hour (up to 90). It gets m(x) = 0.0028x2 +0.28x miles from each gallon of fuel. The highest point on the graph from this function is the point where the tangent is horizontal. Use the derivative to find the speed at which the tractor trailer gets the best gas mileage. What is the gas mileage?
Acceleration • Average acceleration is the rate of change in velocity. • The instantaneous rate of change in velocity is the instantaneous acceleration and is the derivative of velocity.
Finding acceleration example • Find the derivative to find the coordinates. • Use Pythagoreans and trig to find the magnitude and direction of the acceleration
Instantaneous & Average Acceleration • The average acceleration can be found by taking the slope of a velocity time graph. • Instantaneous acceleration can be found by taking the tangent at the point at which the acceleration is desired or finding the limit. Average acceleration =instantaneous acceleration if a is uniform
Acceleration • a is the derivative of v: ex if vx = 40-5t2 then ax = -10t • Ex: if the velocity of a particle varies according to vx = 40-5t2find a over 2 second interval ti = 0 tf= 2 vi=40-5(0)2=40 m/s vf=40-5(2)2=20 m/s • A=20-40 / 2-0 = -10m/s2
Solution continued • Find a at 2 sec t = 2 sec Dt=>0 Dv = vf – vi • vi= 40-5(t)2 • vf = 40-5(t-Dt)2 • Dv = 40-5t2-10tDt-5Dt2 - 40-5t2 substitute • Dv = -10tDt -5Dt2 • Take the derivative to get a= -10t = 10Dt • If a is constant then aavg = alimt=>0 so aavg = -10 t m/s2 or -20 m/s2