Be on time Switch off mobile phones. Put away laptops. Being present = Participating actively

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1 A couple of house rules Be on time Switch off mobile phones Put away laptops Being present = Participating actively

2 Het basisvak Toegepaste Natuurwetenschappen Applied Natural Sciences Leo Pel e mail: 3nab@tue.nl

3 Chapter 2 Motion Along a Straight Line Application of calculus in Mechanics: 1D PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Copyright 212 Pearson Education Inc.

4 LEARNING GOALS How to describe straight-line motion in terms of average velocity, instantaneous velocity, average acceleration, and instantaneous acceleration. How to interpret graphs of position versus time, velocity versus time, and acceleration versus time for straightline motion. How to solve problems involving straight-line motion with constant acceleration, including free-fall problems. How to analyze straight-line motion when the acceleration is not constant.

5 Mechanics ( the limitation ) Each object is considered to be a point mass mass M point mass M book: particle mass M Objects (therefore) do not rotate

position, r, of an object at any instant of time, t.

6 Chapter 2: kinematics Kinematics: the description of motion In 3 dimensions, this means determining the position, r, of an object at any instant of time, t. co-ordinates unit vectors We will limit ourselves to 1D/2D

7 Chapter 2: kinematics in 1D Kinematics: the description of motion Determination of motion of an object along a straight line (1 dimension), means determining the co ordinate, x, of an object at any instant of time, t. x co-ordinate unit vector

8 Motion along a straight line Which quantities of an object do you need to know to be able to describe its motion along a straight line? Answer: velocity and acceleration at any time (as a vector, so including direction!) What data is also needed for a unique determination of position of an object at any time? Answer: Position at t=, and a definition of a co ordinate system (for determination of positive and negative direction)

9 Motion along a line y-axis Position, x, of an object at any time, t (velocity and acceleration can be derived) x 1 at t 1 x-axis Choose co ordinate system Choose x axis along the direction of motion only x co ordinate Choose direction positive x axis Choose origin x= and t= Define reference point : which point represents the position of the object

10 Position as a function of time x(t) y-axis position as a function of time : x(t) x 1 = x(t 1 ) x 2 = x(t 2 ) x-axis average velocity

11 Average Velocity This plot shows the average velocity being measured over shorter and shorter intervals.

12 Average Velocity

13 Instantaneous Velocity Definition: This means that we evaluate the average velocity over a shorter and shorter period of time; as that time becomes infinitesimally small, we have the instantaneous velocity. instantaneous velocity

14 Average velocity Position function average velocity slope (=steepness) of the connecting line between x 1 (=x(t 1 )) en x 2 =x(t 2 )

15 Instantaneous velocity position function position function position function (instantaneous) velocity slope (=steepness) tangent to x(t) Derivate of the position function with respect to time

Motion along a line The graph shows the position functions of two trains running along parallel tracks. Which statement is correct? 1. At t=t B both trains have the same velocity 2.

16 Motion along a line The graph shows the position functions of two trains running along parallel tracks. Which statement is correct? 1. At t=t B both trains have the same velocity 2. Both trains accelerate all the time 3. Both trains have the same velocity at a time instant t<t B 4. Both trains have the same acceleration somewhere on the graph Answer: 3. The slope of curve B is is at a certain time instant t<t B equal to the slope of curve A

17 Velocity function v(t) Velocity function v(t)

18 Acceleration function a(t) Acceleration function a(t) average acceleration (instantaneous) acceleration

should not be confused with the directions of velocity

19 Acceleration function VS Decelartion Acceleration (increasing speed) and deceleration (decreasing speed) should not be confused with the directions of velocity and acceleration: Accelerating Decelerating Decelerating Accelerating

20 Motion with Constant Acceleration Velocity vs. time: Average velocity: Position as a function of time: Velocity as a function of position:

21 Derivation of function v(x) for constant a at v v a v v t a v v a a v v v x x at t v x x 2 ) ( ) ( 2 )2 ( v v v v v a x x ) ( v v a x x

22 Motion with Constant Acceleration The relationship between position and time follows a characteristic curve. Parabola

He applies the brakes and slows with an acceleration of -3.8 m/s 2. (a) If the deer is 2.

23 Example: Hit the Brakes! A park ranger driving at 11.4 m/s in back country suddenly sees a deer frozen in the headlights. He applies the brakes and slows with an acceleration of -3.8 m/s 2. (a) If the deer is 2. m from the ranger s car when the brakes are applied, how close does the ranger come to hitting the deer? (b) What is the stopping time? v v () (11.4 m/s) x 17.1 m 2 2a 2( 3.8 m/s ) v (11.4 m/s) vv at t 3. s 2 a ( 3.8 m/s ) d 2. m 17.1 m 2.9 m

24 Free fall A strobe light begins to fire as the apple is dropped. Notice how the space between images increases as the bal s velocity grows.

25 Free fall g = 9.8 m/s 2 ~15 3 km/h

26 Free fall Aristotle thought that heavier bodies would fall faster. Galileo is said to have dropped two objects, one light and one heavy, from the top of the Leaning Tower of Pisa to test his assertion that all bodies fall at the same rate.

27 Apollo

Motion along a line The graph shows the position function of a train that moves along a straight track. Which statement is correct? 1. The train moves with constant velocity 2.

28 Motion along a line The graph shows the position function of a train that moves along a straight track. Which statement is correct? 1. The train moves with constant velocity 2. The train decelerated all the time 3. The train accelerates the all the time 4. The train accelerates and after that it decelerates Answer: 2. The slope of the curve indicates the velocity; the slope, and hence the velocity, decreases all the time => the train decelerates.

29 Speed of a Lava Bomb A volcano shoots out blobs of molten lava (lava bombs) from its summit. A geologist observing the eruption uses a stopwatch to time the flight of a particular lava bomb that is projected straight upward. If the time for it to rise and fall back to its launch height is 4.75 s, what is its initial speed and how high did it go? (Use g =9.81m/s 2.) x x v t gt x t( v gt) Either t or v gt v 1 2 gt (9.81 m/s )(4.75 s) 23.3 m/s At maximum height, v v 2gx v (23.3 m/s) x 2 2g 2(9.81 m/s ) 27.7 m

30 Summary From the position function, x(t), one can obtain the velocity function, v(t), and the acceleration function, a(t), via differentiation with respect to time xt () vt () dx at () dv d x dt dt dt 2 2 If a(t) is known, can v(t) and x(t) be determined?

31 Suppose a(t) is known! at () dv dt C vt () atdt () v vt () dx dt C xt () vtdt () x If the acceleration function, a(t), is known, then the velocity function, v(t), can be obtained by integration and from another integration step the position function, x(t), can be determined if C v en C x are known

32 Determine v(t) and x(t) if a(t) is known Initial values of v(t) and x(t) have to be known vt () vt ( ) at () dt xt () xt ( ) vt () dt t t t t Apply to: constant acceleration, i.e. a(t)=a =constant

a(t), v(t) en x(t) van een valbeweging at

33 a(t), v(t) en x(t) van een valbeweging at () g9.8(1)m/s vt ( ) v; xt ( ) x 2 Constant acceleration Initial values x 9.8 m/s 2 t 1 1 vt ( ) v() at ( ) dtv g dt v g t 1 1 t t1 t xt ( ) x() vt ( ) dt x ( v g t) dt x v t g t Area under the curve

34 Constant acceleration Motion for which the acceleration is constant a(t) = a v(t) = v + a t x(t) = x + v t + ½ a t 2 v initial velocity x initial position

35 Summary One dimensional motion object: x(t), v(t), a(t) x( t) dx v( t) a( t) dt dv dt differentiation v( t) x( t) v( t x( t ) ) t t t t a( t) dt v( t) dt Integration Initial values needed Motion with constant acceleration: a = constant Acceleration: free fall (a=g)

36 Free fall A ball is thrown straight upwards from a height, h, with a velocity, v, and hits the ground at velocity v 1. What is the velocity of the ball when hitting the ground when the ball is thrown from the same height, h, but now straight downwards with velocity v, (ignore air resistence). 1. The velocity is larger than v 1 2. The velocity is equal to v 1 3. The velocity is smaller than v 1 4. Not enough data to know Answer: 2. In the former situation, when coming back at height h, the ball will have a velocity equal to the velocity with which is was thrown upwards, but now pointing downwards

37 v(t) = v -a t V o v initial velocity x(t) = + v t - ½ a t 2 initial position Back at initial position as: v t - ½ a t 2 = h V o V? t( v -½ a t)= t= or t=2v / a v(t) = v -a 2v / a =-v So same final velocity

38 Summary

39 Summary

General Physics (PHY 170) Chap 2. Acceleration motion with constant acceleration. Tuesday, January 15, 13

General Physics (PHY 170) Chap 2 Acceleration motion with constant acceleration 1 Average Acceleration Changing velocity (non-uniform) means an acceleration is present Average acceleration is the rate