Lecture. Chapter : Motion along a Straight Line Displacement, Time, Velocity 3/6/05
One-Dimensional Motion The area of physics that we focus on is called mechanics: the study of the relationships between force, matter and motion For now we focus on kinematics: the language used to describe motion Later we will study dynamics: the relationship between motion and its causes (forces) Simplest kind of motion: -D motion (along a straight line) A particle is a model of moing body in absence of effects such as change of shape and rotation Velocity and acceleration are physical quantities to describe the motion of particle Velocity and acceleration are ectors 3/6/05
Position and Displacement Motion is purely translational, when there is no rotation inoled. Any object that is undergoing purely translational motion can be described as a point particle (an object with no size). The position of a particle is a ector that points from the origin of a coordinate system to the location of the particle The displacement of a particle oer a gien time interal is a ector that points from its initial position to its final position. It is the change in position of the particle. To study the motion, we need coordinate system 3/6/05 3
Position and Displacement Motion of the particle on the dragster can be described in terms of the change in particle s position oer time interal Displacement of particle is a ector pointing from P to P along the -ais 3/6/05 4
5 3/6/05 Aerage Velocity Aerage elocity during this time interal is a ector quantity whose -component is the change in diided by the time interal t t t a t t t
Aerage Velocity Aerage elocity is positie when during the time interal coordinate increased and particle moed in the positie direction If particle moes in negatie -direction during time interal, aerage elocity is negatie 9m 77m 58m t t t 5.0s 6.0s 9. 0s a t 58m 9m / 9.0s s 3/6/05 6 6
X-t Graph This graph is pictorial way to represent how particle position changes in time Aerage elocity depends only on total displacement, not on the details of what happens during time interal t The aerage speed of a particle is scalar quantity that is equal to the total distance traeled diided by the total time elapsed. 3/6/05 7
Aerage Velocity 3/6/05 8
Instantaneous Velocity Instantaneous elocity of a particle is a ector equal to the limit of the aerage elocity as the time interal approaches zero. It equals the instantaneous rate of change of position with respect to time. lim t 0 t d dt 3/6/05 9
Instantaneous Velocity On a graph of position as a function of time for one-dimensional motion, the instantaneous elocity at a point is equal to the slope of the tangent to the cure at that point. 3/6/05 0
Instantaneous Velocity 3/6/05
Instantaneous Velocity Concept Question The graph shows position ersus time for a particle undergoing -D motion. At which point(s) is the elocity positie? At which point(s) is the elocity negatie? At which point(s) is the elocity zero? At which point is speed the greatest? 3/6/05
Acceleration Acceleration If the elocity of an object is changing with time, then the object is undergoing an acceleration. Acceleration is a measure of the rate of change of elocity with respect to time. Acceleration is a ector quantity. In straight-line motion its only non-zero component is along the ais along which the motion takes place. 3/6/05 3
Aerage Acceleration Aerage Acceleration oer a gien time interal is defined as the change in elocity diided by the change in time. In SI units acceleration has units of m/s. a a t t t 3/6/05 4
Instantaneous Acceleration Instantaneous acceleration of an object is obtained by letting the time interal in the aboe definition of aerage acceleration become ery small. Specifically, the instantaneous acceleration is the limit of the aerage acceleration as the time interal approaches zero: a lim t 0 t d dt 3/6/05 5
Acceleration of Graphs 3/6/05 6
Acceleration of Graphs 3/6/05 7
Acceleration of Graphs 3/6/05 8
Constant Acceleration Motion In the special case of constant acceleration: the elocity will be a linear function of time, and the position will be a quadratic function of time. For this type of motion, the relationships between position, elocity and acceleration take on the simple forms : a t t a t 0 0 0 a t 3/6/05 9
Constant Acceleration Motion 3/6/05 0
3/6/05 Constant Acceleration Motion Position of a particle moing with constant acceleration 0 0 t t a 0 a t a 0 t a t a a 0 0 0 t t a 0 0 0 0 t a t
3/6/05 Constant Acceleration Motion Relationship between position of a particle moing with constant acceleration, and elocity and acceleration itself: 0 0 t a t t a 0 a t 0 0 0 0 0 a a a ) ( 0 0 a
A car is moing along the -ais. The graph below shows the position of the object as function of time. The questions below refer to that object. Find the displacement between t=0s and t=.0s Find the displacement between t=0s and t=4.0s Find the distance coered between t=0s and t=4.0s Find the acceleration at t=3.0s Find the distance coered between t=3.0s and t=7.0s Find the instantaneous elocity at t=3.0s Find the instantaneous elocity at t=4.5s Find the aerage elocity between t=.0s and t=6.0s 3/6/05 3
A car is moing along the -ais. The graph below shows the elocity of the object as function of time. The questions below refer to that object. Find the elocity at t=3s Find the displacement between t=0s and t=4.0s Find the aerage elocity between t=0s and t=4.0s Find the aerage speed between t=0s and t=4.0s Find the acceleration at t=3.0s Find the distance coered between t=0s and t=7.0s Find the speed at t=4.0s 3/6/05 4
Freely Falling Bodies The constant acceleration of a freely falling body is called acceleration due to graity, g Approimate alue near earth s surface g = 9.8 m/s = 980 cm/s = 3 ft/s g is the magnitude of a ector, it is always positie number Acceleration due to graity Near the sun: 70 m/s Near the moon:.6 m/s Eact g alue aries with location 3/6/05 5
Problem An egg is thrown ertically upward from a point near the cornice of a tall building. It just misses the cornice on the way down and passes a point 50 m below its starting point 5.0 s after it leaes the thrower s hand. Ignore air resistance. a. What is the initial speed of the egg? b. How high does it rise aboe its starting point? c. What is the magnitude of its elocity at its highest point? d. What are the magnitude and direction of its acceleration at the highest point? 3/6/05 6
Step : Draw it h, height aboe building 50 m in 5 s 3/6/05 7
Make some definitions Let t=5 s Let o = initial elocity, in positie direction Let g=9.8 m/s, acceleration due to graity, in negatie direction. total distance =-50 m () t at o y() t at ot yo f 0 a( y f yo) 3/6/05 8
Apply Definitions and Sole Let t=5 s Let o = initial elocity, in positie direction Let g=9.8 m/s, acceleration due to graity, in negatie direction. This will be a total distance =-50 m =y(t)-y o y() t at ot yo y() t yo at ot 50 g(5) o (5) 7.5 5.5 0 4.5 m / s 0 3/6/05 9
Part B) Find height aboe bldg 0 =4.5 m/s At top of the arc, f =0 At t=0, let y 0 =0 () t at o y() t at ot yo f 0 a( y f yo) 3/6/05 30
Part B) Find height aboe bldg 0 =4.5 m/s At top of the arc, f =0 At t=0, let y 0 =0 Let y f =h f 0 a( y f yo) g( h 0) f 0 0 4.5 g 0.7 m h h 3/6/05 3
Parts C) and D) At the top of the arc, =0 m/s Always in this problem a=-g=-9.8 m/s 3/6/05 3
Problem A car 3.5 m in length and traeling at a constant speed of 0 m/s is approaching an intersection which is 0 m wide. The light turns yellow when the front of the car is 50 m from the beginning of the intersection. If the drier steps on the brake, the car will slow at -3.8 m/s. If the drier steps on the gas pedal, the car will accelerate at.3 m/s. The light will be yellow for 3 seconds. Ignore reaction time. To aoid being in the intersection when the light changes to red, accelerate or brake? 3/6/05 33
Step : Draw It! 3.5 m 50 m In order for the car to run the green light, It must traerse 3.5+50+0 m=73.5 m so that no part of the car is in the intersection 0 m Braking is easier, it must traerse less than 50 m 3/6/05 34
Our Options We know that t=3 s a is either.3 m/s or - 3.8 m/s o =0 m/s Total distance is either 73.5 m or 50 m () t at o () t at ot o f 0 a( f o ) 3/6/05 35
Our Options We know that t=3 s a is either.3 m/s or -3.8 m/s o =0 m/s Total distance is either >73.5 m or <50 m () t at ot o Braking () t at ot t ( t) 4.9 m ( ) o (3.8) 3 03 o Run It! t ( t) 70.4 m () t at ot o ( ) o (.3) 3 03 o 3/6/05 36
Problem A physics student with too much free time drops a water melon from the roof of a building. He hears the sound of the watermelon going splat.5 s later. How high is the building? The speed of sound is 340 m/s and ignore air resistance. 3/6/05 37
Step : Draw It! h Splat! 3/6/05 38
Some Hard Thinkin The melon eperiences an acceleration due to graity. The student merely dropped it, so its initial elocity was 0. The sound wae is unaffected by graity so it moes with constant elocity from the ground toward the student. These are two separate eents with a total time of.5 s 3/6/05 39
Some Hard Thinkin part The equation for the distance that the melon traerses is y=-/*g*(t ) where y= height of bldg and t is the time for the fall. The equation of distance for the sound wae is y= s *t where s = speed of sound =340 m/s The total time for all this to transpire is.5 s or.5 s =t +t 3/6/05 40
Soling y gt y 4.9t y 340t.5 t t t.5 t y 340*.5 t 850 340t 4.9t 850 340 t or 4.9t 340t 850 0 t t y or t b b 4ac a 4.9.46 or.4 s 4.9*.4 8.6 m.5.46 0.084 y 340*0.084 8.6 m 340 340 4 4.9 ( 850) 3/6/05 4
Problem.89 A painter is standing on scaffolding that is raised at a constant speed. As he traels upward, he accidentally nudges a paint can off the scaffolding and it falls 5 m to the ground. You are watching and measure with your stopwatch that it takes 3.5 s for the can to reach the ground. a. What is the speed of the can just before it hits the ground? b. Another painter is standing on a ledge with his hands 4 m aboe the can when it falls. He has lightningfast reflees and can catch if at all possible. Does he get a chance? 3/6/05 4
Part a) Make some definitions Let t=3.5 s Let o = initial elocity, in positie direction Let g=9.8 m/s, acceleration due to graity, in negatie direction. total distance =-5 m () t at o y() t at ot yo f 0 a( y f yo) 3/6/05 43
Make some definitions We must sole this equation for () t at o o And then we must use this equation to sole for the elocity y() t at ot yo f 0 a( y f yo) 3/6/05 44
Soling y( t) y 5 y() t at ot yo o 36.75 3.5 5 g(3.5) o 3.5 o.3 m / s o () t at o ( t) g *3.5.3 ( t) 0.5 m 3/6/05 45
Part B) Make some definitions The word falls is slightly misleading, the can first rises in the air and then falls to the ground. What it is asking for is how high the can flies aboe the release point; it must be greater than 4 m So we are soling for a total y displacement (y f -y o ) At the top of the arc, f =0 Let o = initial elocity, in positie direction,.3 m/s Let g=9.8 m/s, acceleration due to graity, in negatie direction. () t at o y() t at ot yo f 0 a( y f yo) 3/6/05 46
Part B) Make some definitions What it is asking for is how high the can flies aboe the release point; it must be greater than 4 m So we are soling for a total y displacement (y f -y o ) At the top of the arc, f =0 Let o = initial elocity, in positie direction,.3 m/s Let g=9.8 m/s, acceleration due to graity, in negatie direction. f 0 a( y f yo) 0.3 g( y f yo).3 g ( y y ) 6.5 m ( y y ) Yes, he can catch it f f o o 3/6/05 47