Relative Motion Notes Key PDF

2.1 Learning Targets  I can analyse motion from different frames of Relative Motion – reference....

2.1 Learning Targets  I can analyse motion from different frames of Relative Motion – reference.  I can draw vector diagrams. 1D Relative Motion  I can perform relative motion calculations. Relative Motion Learning Check Relative Velocity: the description of the velocity of one object You are standing at rest at a SkyTrain relative (compared) to another. platform. The SkyTrain enters the station traveling at 30 km/h East relative to you on Reference frame: a set of coordinates that can be used to the platform. What is your velocity relative determine positions and velocities of objects in that frame. to a passenger sitting inside the SkyTrain? In most examples we have examined so far, this reference frame a. 0 km/h has been Earth (aka the ground). If you say a person is sitting in a b. 30 km/h West train moving at 10 m/s East, then you imply the person on the train c. 30 km/h East is moving relative to the surface of Earth at this velocity (Earth is d. 60 km/h East the reference frame and considered to be at rest). Example 1: A man walks to the right with a a) What is the person’s velocity relative to the platform? velocity of 2 m/s on a platform that moves with a velocity of 1 m/s to the right. Platform Reference frame: ____________ 2 vperson-platform = __________m/s 2 m/ s object b) WhatSrf is the person’s velocity relative to the ground? Ground Reference frame: ____________ vperson-ground = vperson-platform + vplatform-ground IT 1 m/ s vperson-ground = _____m/s + _____ m/s vperson-ground = _____ m/s Adding Vectors with Relative Motion When trying to observe motion from different reference frames (like we saw in Example 1, part b), we use vector addition and vector diagrams. Review: Drawing Vector Diagrams Vectors have both magnitude and direction. To show vectors… 1. Draw an arrow in the direction of the vector. tip 2. Scale the length to represent the vector’s magnitude. When we are adding vectors, we use the tip-to-tail (or head-to- tail) method. Adding by the tip-to-tail method means to move one vector so that its tail lies on the tip of the first vector. The resultant vector (the sum of the vectors) is simply drawn from the tail of the first vector to the tip of the last vector. Example 1 cont’d Draw the vector diagram for the person’s velocity relative to the ground. vperson-ground = vperson-platform + vplatform-ground refutant per plat Fp to tail Vplat Vper gr g  Va o Learning Check Subtracting Vectors with Relative Motion Two cars, Y and X, are travelling East When trying to observe motion from difference along Scott Road. Car Y is travelling at reference frames we may also sometimes use 4.0 m/s and Car X is travelling at 6.0 vector subtraction. m/s. How fast is Car Y moving relative Eg. What is the velocity of person A relative to to Car X? person B? vA-Ground = vA-B + vB-Ground  vA-B = vA-Groundth - vVo a. 4.0 m/s East Ground B-Ground b. 6.0 m/s West c. 10 m/s East To subtract vectors, we use the tip to tail method, but we simply add the d. 2.0 m/s West opposite (in direction) of the vector being subtracted. VA-Ground -VB-Ground VA-Ground - VB- Ground = + VA-Ground -VB-Ground VA-B Learning Check cont’d Draw the vector diagram for the velocity of Car Y relative to Car X. Ground around Vyground Vy VxGround Vy x VyGround tf Kground Nyx Vy Ground C Vxground Example 2: Let’s say you are standing on the back of a pickup truck that is moving at 20 m/s relative to the ground and you are throwing a ball backwards. You know that you can throw the ball away from you at exactly 15 m/s every time. a. If a person were standing on the sidewalk, how fast would they say the ball is moving? Show a vector diagram. b. If the person began to run towards the truck at 2 m/s, how fast would they say the truck is moving? Show a vector diagram. Vbanground banneck 15m15 20m15 truckground 9 Vban ground Vban truck truckground U Vio g Ismls 20m15 t Smls forwards Vt G b VtruckgroundVtruckperson Vpersonground truckperson Vtruck ground Vpersonground Vt G VP g 20m s f 2m15 Vt p 22m15 forwards 2.2 Learning Targets  I can analyse motion from different frames of Relative Motion - reference.  I can draw vector diagrams. 2D Relative Motion  I can perform relative motion calculations. Example 1: Crossing a River A boat can travel 2.30 m/s in water. It crosses a river with a current of 1.50 m/s West. a) If the boat heads directly North across the river, what is the velocity of the boat relative to the shore? b) What direction must you point the boat to land directly across the river (North)? MEEEE É ant Nwstymls m's 1 13 Ves 2.75ms j boatwater watershore i I 0 33TWofN b Vw s Vo n yo the'Voys Vow Vb s Vw s Vow Vbs Vws I V8S f Vo w 2.30 Sino 5 0 0 41 EAN Component Method (can be used for adding more than 2 vectors) The steps to follow this method are: 1. Draw each vector. 2. Break down vectors into their x and y components (check the sign for direction!) 3. Add the x components 4. Add the y components 5. Use the Pythagorean theorem and trigonometry to get the magnitude and direction of the resultant vector Example 3: Airplane and Wind (Component) Vanewind Vwindground Vpianeground An airplane heading at 450 km/h in a direction of 30O N of E encounters a 75 km/h wind blowing towards a direction of 50O W of N. What is the resultant velocity of the airplane relative to the ground? Airplane vector: Wind vector: x-component: x-component: Vpwx VpwCos0 Vwox Vwo.si 7Ssin5 389.71 yp7Vpwx 4500530 Vwox 13.0 PWYVPwx Éyjwa Vox 57.45 Ywest Vpwx y-component: y-component: Vpwy Vpwsin Voy VwocosO Vpwy 450sin30 V07 7500550 Vway 48.21 Vpwy 225 Adding the two vectors: Total resultant: x-components of resultant: Ypg VpwxtVwGx YPG 273.21 pox 389.71 5745 10 Vox 1332.26 up 332.26 Vp 332.265 1273.21 y-components of resultant: Vpoy VpwytVwgy Vp 430km h 39 NofE Vpoy 225 48.21 Otan 333 14 Upon 273.21 north Trig Method Use the tip to tail method to draw the vector addition diagram. Use a combination of the sine law and/or cosine law to calculate the magnitude and direction of the resultant vector. Example 3: Airplane and Wind (Trig) An airplane heading at 450 km/h in a direction of 30O N of E encounters a 75 km/h wind blowing towards a direction of 50O W of N. What is the resultant velocity of the airplane relative to the ground? plane wind Vwind Vplane ground ground 4MW G 60 1418980 70 Vp G 14 Vp W join is Vp62 Vp w Vw o 2vpwvw o cos0 sying.SI 0 9 Vpof 45072 175 214507175005700,90 30 39 Note Vp 430km h 2.3 Learning Targets  I understand the special theory of relativity and can Relative Motion – describe its implications. Special Relativity  I can perform special relativity calculations. Special Theory of Relativity As previously discuss, Maxwell’s equations were not compatible with Newtonian mechanics (which worked within the frames of Galilean relativity). This frustrated Albert Einstein, which led to his proposal (1903) of the special theory of relativity. → Special relativity corrects mechanics to handle situations involving ALL motions - especially those at speeds close to the speed of light! Special relativity is an explanation on how speeds (especially those that are very close to the speed of light) affect mass, time, and space. Two Postulates of Special Relativity Relativistic Effect – Time Dilation Einstein suggested that absolute Imagine a light clock inside a moving spaceship, there are two observers. motion has no meaning, that all Observer #1 (inside Observer #2 (on Earth) that motion is relative. The special theory the spaceship) sees sees the clock as moving. of relativity is based on two fundamental assumptions that he the clock as stationary. relativistictime t propertime made (based on previously established knowledge). These fundamental assumptions are called to postulates. u s 1. The laws of physics are true in all inertial (non-accelerating) frames of reference. Light travels a longer distance in the moving frame of reference than 2. The speed of light is the same for in the stationary frame. If the speed of light is the same in all all observers, regardless of the reference frames and v = d/t, then a longer distance must take more time! YETI motion of the light source or the observer. According to Einstein’s Special Theory of Relativity, a clock that is moving will run slow. This ‘stretching’ of time by a moving object is called time dilation. Calculating Time Dilation Lorentz Factor (ϒ) 8 The factor by which time is t ft dilated for an object while that object is moving. t = Time measured by an observer where both events occur at different locations (relativistic time) to = Time measured by an observer where both events occur at the same location (proper time) v = speed of travel c = speed of light (3.0 × 108 m/s)  85 oflight speed Learning Check Example 1: A certain isotope has a half-life of 5.0 s. a) If it is fired into a particle accelerator and reaches a speed of 0.85c, what is A comet travels in space the Lorentz factor? and glows every 10 t b) What will the new half-life be at this speed? seconds. You observe this HEE HEE comet through a telescope on Earth and see it glow a every 11.5 seconds. Who has measured the Y Fg proper time? 1.89 a) You b) The Comet c) d) Both Neither b t Jt 4.8915.0s 9Ss Relativistic Effect - Length Contraction Calculating Length Contraction One thing all observers agree upon is relative speed. Lf Since v = d/t, this implies that distance, too, depends on the observer’s relative motion. If two observers can measure different times, then they must also measure different distances for speed to remain the same. L = the distance between two points measured by an When an object is in motion, its measured length observer who is in motion relative to both points shrinks in its direction of motion. This is called length (relativistic length) contraction. Lo = the distance between two points measured by an observer who is at rest relative to both points (proper length) v = speed of travel c = speed of light Learning Check Lofroperlength Learning Check y A spaceship has left Earth and is travelling towards the A spaceship has left Earth and is travelling towards the star Alpha Centauri. You observe the spaceship through star Alpha Centauri. You now measure the distance a telescope and measure its length to be 100 meters. between Earth and Alpha Centauri. You measure this The astronauts inside the spaceship measure it length distance to be 4.367 lightyears. The astronauts also to be 120 meters. Who has measured the proper make the same measurement. Will their measured length? distance be…. Lrelativistic length a) You a) Equal to 4.367 lightyears b) The astronauts b) Greater than 4.367 lightyears c) Both c) Less than 4.367 lightyears d) Neither d) Not enough information Example 2: A football is 25cm long. It is travelling at 60% the speed of light (nice toss!) How long would the football appear to be for the person receiving the pass? GO60c L L L VIT 25cmV1 1 91 25cm VT0TO 20cm Mass-Energy Equivalence Relativistic Effect - Relativistic Mass As an object increase in To a stationary observer, the mass of a particle travelling at extremely high speed, it also gains mass. velocities becomes heavier. This is due to the mass- m = mass measured by an observer who is moving energy equivalence. relative to the object (relativistic mass) Simply put, mass is another mo = mass measured by an observer who is form of energy. Energy can stationary relative to the object (rest mass) turn into mass and mass v = speed of travel can turn into energy. c = speed of light M J Mo The conversion between matter and energy can be Relativistic mass also demonstrates why it is impossible for objects to travel at the quantified with Einstein’s speed of light. As an object speeds up, it requires more and more energy to most famous equation: accelerate as it gains more and more mass. The energy required to accelerate a body to the speed of light is infinitely large since the mass of the object would also E = mc2! be infinitely large. Example 3: A pie with a rest mass of 4.0 kg is thrown at your teacher’s face at an unknown speed. Ouch! According to her calculations the mass of the pie when it hits her face is 5.6 kg. What is the speed of the pie (expressed as a percentage of the speed of light)? m m FE III Tm t t Gm E HIT ETHETVHIT o.i v o.k Learning Check: The following diagram shows a spaceship travelling to the star Alpha Centauri in two different reference frames (observer on Earth and observer in ship). Using your understanding of time dilation, length contraction and relativistic mass, label the following variables on the diagram. Proper Time (to) Relativistic Time (t) Proper Length (Lo) Relativistic Length (L) Rest Mass (mo) Relativistic Mass (m) t m Lo to Mo L