PHYS 1443 Section 004 Lecture #4 Thursday, Sept. 4, 014 One Dimensional Motion Motion under constant acceleration One dimensional Kinematic Equations How do we sole kinematic problems? Falling motions Motion in two dimensions Coordinate system Vector and scalars, their operations Today s homework is homework #3, due 11pm, Thursday, Sept. 11!! 1
Quiz # Announcements Beginning of the class coming Thursday, Sept. 11 Coers CH1.1 through what we learn Tuesday, Sept. 9 Miture of multiple choice and free response problems Bring your calculator but DO NOT input formula into it! Your phones or portable computers are NOT allowed as a replacement! You can prepare a one 8.511.5 sheet (front and back) of handwritten formulae and alues of constants for the eam None of the parts of the solutions of any problems No deried formulae, deriations of equations or word definitions! No additional formulae or alues of constants will be proided! First term eam moed from Tuesday, Sept. 3 to Thursday, Sept. 5! Please make a note!
Special Project # for Etra Credit Show that the trajectory of a projectile motion is a parabola!! 0 points Due: Thursday, Sept. 11 You MUST show full details of your OWN computations to obtain any credit Beyond what was coered in this lecture note and in the book! 3
Displacement, Velocity, Speed & Acceleration Displacement Aerage elocity Aerage speed Instantaneous elocity Instantaneous speed Aerage acceleration Instantaneous acceleration Δ lim 0 Thursday, Sept. 4, 014 Δt 0 PHYS 1443-004, Fall 014 4 t f f f i ti i Δ Δt Δt Δ Δt Total Distance Traeled Total Time Spent lim 0 Δ Δt Δt a a t f f lim ti i Δ Δt d dt d dt Δ Δt d dt d d dt dt d dt
Eample for Acceleration Velocity,, is epress in: (t) Find the aerage acceleration in time interal, t0 to t.0s Instantaneous Acceleration at any time i (t i 0) f (t f.0) a Find instantaneous acceleration at any time t and t.0s a ( t) 40(m / s) f i t f t i Δ Δt d dt d dt ( 40 5.0 ) ( 40 5t ) ( ) 10t m s ( 40 5t )m / s 0(m / s) 0 40.0 0 10(m / s ) Instantaneous Acceleration at any time t.0s a(t.0) 10 (.0) 0(m / s ) 5
Eample for Acceleration Position is epress in: Find the particle s elocity function (t) and the acceleration function a(t). (t) d dt d dt a (t) d dt 4 7t + t 3 ( m) ( 4 7t + t 3 ) d dt d dt Find the aerage acceleration between t.0s and t4.0s (t ) 7 + 3t ( 7 + 3t ) (t 4) 7 + 3t 7 + 3 4 a (t 4) (t ) 4 7 + 3 7 + 3t ( m s) +6t ( m s ) Find the aerage elocity between t.0s and t4.0s ( ) 15( m s) ( ) +1( m s) ( ) 36 1 15 +18( m ) s 6
Check point on Acceleration Determine whether each of the following statements is correct! When an object is moing in a constant elocity ( 0 ), there is no acceleration (a0) Correct! The elocity does not change as a function of time. There is no acceleration when an object is not moing! When an object is moing with an increasing speed, the sign of the acceleration is always positie (a>0). Incorrect! The sign is negatie if the object is moing in negatie direction! When an object is moing with a decreasing speed, the sign of the acceleration is always negatie (a<0) Incorrect! The sign is positie if the object is moing in negatie direction! In all cases, the sign of the elocity is always positie, unless the direction of the motion changes. Incorrect! The sign depends on the direction of the motion. Is there any acceleration if an object moes in a constant speed but changes its direction? The answer is YES!! 7
One Dimensional Motion Let s focus on the simplest case: acceleration is a constant (aa 0 ) Using the definitions of aerage acceleration and elocity, we can derie equations of motion (description of motion, elocity and position as a function of time) a a f f i f i (If t f t and t i 0) a f i For constant acceleration, aerage elocity is a simple numeric aerage t f f t ti i t (If t f t and t i 0) Resulting Equation of Motion becomes f i i f + t t f i + t i + at f i i i + + at + 1 t 1 i + it+ at at 8
Kinematic Equations of Motion on a Straight Line Under Constant Acceleration f ( t) i + at Velocity as a function of time f i 1 t 1 ( f + i )t Displacement as a function of elocities and time f i + it + 1 at Displacement as a function of time, elocity, and acceleration f i + ( a f ) i Velocity as a function of Displacement and acceleration You may use different forms of Kinematic equations, depending on the information gien to you in specific physical problems!! 9
How do we sole a problem using a kinematic formula under constant acceleration? Identify what information is gien in the problem. Initial and final elocity? Acceleration? Distance, initial position or final position? Time? Conert the units of all quantities to SI units to be consistent. Identify what the problem wants Identify which kinematic formula is most appropriate and easiest to sole for what the problem wants. Frequently multiple formulae can gie you the answer for the quantity you are looking for. è Do not just use any formula but use the one that can be easiest to sole. Sole the equations for the quantity or quantities wanted. 10
f f Eample Suppose you want to design an air-bag system that can protect the drier in a headon collision at a speed 100km/hr (~60miles/hr). Estimate how fast the air-bag must inflate to effectiely protect the drier. Assume the car crumples upon impact oer a distance of about 1m. How does the use of a seat belt help the drier? How long does it take for the car to come to a full stop? The initial speed of the car is We also know that Using the kinematic formula The acceleration is a Thus the time for air-bag to deploy is i 0 m/ s i f ( f i) and t f i As long as it takes for it to crumple. 100 km / h 100000 m 8 m/ s 3600s 1m i + a f i ( ) 0 8 m/ s 1m f a ( ) i 390 m/ s 0 8 m/ s 390 m/ s 0.07s 11
Check point for conceptual understanding Which of the following equations for positions of a particle as a function of time can the four kinematic equations applicable? (a) (b) (c) (d) 3t 4 5t 3 + 4t + 6 t 4 t 5t 3 What is the key here? Finding which equation gies a constant acceleration using its definition! Yes, you are right! The answers are (a) and (d)! 1
Falling Motion Falling motion is a motion under the influence of the graitational pull (graity) only; Which direction is a freely falling object moing? Yes, down to the center of the earth!! A motion under constant acceleration All kinematic formula we learned can be used to sole for falling motions. Graitational acceleration is inersely proportional to the distance between the object and the center of the earth The magnitude of the graitational acceleration is g9.80m/s on the surface of the earth, most of the time. The direction of graitational acceleration is ALWAYS toward the center of the earth, which we normally call (-y); where up and down direction are indicated as the ariable y Thus the correct denotation of graitational acceleration on the surface of the earth is g-9.80m/s when +y points upward 13
Eample for Using 1D Kinematic Equations on a Falling object Stone was thrown straight upward at t0 with +0.0m/s initial elocity on the roof of a 50.0m tall building, What is the acceleration in this motion? f g-9.80m/s (a) Find the time the stone reaches at the maimum height. What is so special about the maimum height? V0 yi ayt + + 0.0 9.80t 0.00 m/ s Sole for t (b) Find the maimum height. t 0.0 9.80.04s 1 1 y f yi+ yit+ at y 50.0 + 0.04 + ( 9.80) (.04) 50.0 + 0.4 70.4( m) 14
Eample of a Falling Object cnt d (c) Find the time the stone reaches back to its original height. yf t.04 4.08s (d) Find the elocity of the stone when it reaches its original height. yi + 0.0( m/ s) + ayt 0.0 ( 9.80) 4.08 (e) Find the elocity and position of the stone at t5.00s. Velocity Position yf y f + ayt 0.0 + ( 9.80) 5.00 9.0( m/ s) yi 1 yi+ yit+ at y 50.0 + 0.0 5.00 + 1 ( 9.80) (5.00) + 7.5( m) 15