Kinematics Varying Accelerations (1D) Vectors (2D)

Transcription

1 Kinematics Varying Accelerations (1D) Vectors (2D) Lana heridan De Anza College ept 29, 2017

2 Last time kinematic equations using kinematic equations

3 Overview falling objects and g varying acceleration vectors components vector addition and subtraction

4 Falling Objects One common scenario of interest where acceleration is constant is falling objects. Objects (with mass) near the Earth s surface accelerate at a constant rate of g = 9.8 ms 2. (Or, about 10 ms 2 ) The kinematics equations for constant acceleration all apply.

5 Falling Objects One common scenario of interest where acceleration is constant is falling objects. Objects (with mass) near the Earth s surface accelerate at a constant rate of g = 9.8 ms 2. (Or, about 10 ms 2 ) The kinematics equations for constant acceleration all apply. Example You drop a rock off of a 50 m tall building. With what velocity does it strike the ground? How long does it take to fall to the ground?

6 Falling Objects Example You drop a rock off of a 50 m tall building. With what velocity does it strike the ground? How long does it take to fall to the ground? What quantities do we know? Which ones do we want to predict?

7 Falling Objects Example You drop a rock off of a 50 m tall building. With what velocity does it strike the ground? How long does it take to fall to the ground? What quantities do we know? Which ones do we want to predict? Which equation(s) should we use?

8 Falling Objects Example You drop a rock off of a 50 m tall building. With what velocity does it strike the ground? How long does it take to fall to the ground? What quantities do we know? Which ones do we want to predict? Which equation(s) should we use?

9 Falling Objects CHAPTER 2 KINEMATIC 65 Figure 2.40 Vertical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearly with time and that acceleration is constant. Misconception Alert! Notice that the position vs. time graph shows vertical position only. It is easy to get the impression that the graph shows some horizontal motion the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is time, not space. The 1 Opentax Physics

10 Acceleration in terms of g Because g is a constant, and because we have a good intuition for it, we can use it as a unit of acceleration. ( How many times g is this acceleration? )

11 Acceleration in terms of g Because g is a constant, and because we have a good intuition for it, we can use it as a unit of acceleration. ( How many times g is this acceleration? ) Consider the drag race car in the earlier example. For the car the maximum acceleration was a = 26.0 m s 2. How many gs is that? A 0 B roughly 1 C roughly 2 and a half D 26

12 Acceleration of a Falling Object Question A baseball is thrown straight up. It reaches a peak height of 15 m, measured from the ground, in a time 1.7 s. Treating up as the positive direction, what is the acceleration of the ball when it reached its peak height? A 0 m/s 2 B 8.8 m/s 2 C +8.8 m/s 2 D 9.8 m/s 2 1 Princeton Review: Cracking the AP Physics Exam

13 1-D Kinematics with varying acceleration If we have a varying acceleration a(t), we should use calculus not the kinematics equations. In the homework (question 57), there is a situation where the jerk, J = da dt is constant, but not zero. In that case the acceleration is not constant: so, a = t 0 J dt a(t) = a i + Jt The velocity cannot be found by using v(t) = v i + at! Instead, integrate again.

14 1-D Kinematics with varying acceleration If we have a varying acceleration a(t), we should use calculus not the kinematics equations. In the homework (question 57), there is a situation where the jerk, J = da dt is constant, but not zero. In that case the acceleration is not constant: a = t 0 J dt so, a(t) = a i + Jt The velocity cannot be found by using v(t) = v i + at! Instead, integrate again.

15 1-D Kinematics with varying acceleration uppose you have a particle with a varying acceleration, but you don t know a as a function of t. Instead, you know a as a function of position, x: a(x) Can you use that knowledge to find changes in velocity, displacement, etc?

16 1-D Kinematics with varying acceleration uppose you have a particle with a varying acceleration, but you don t know a as a function of t. Instead, you know a as a function of position, x: a(x) Can you use that knowledge to find changes in velocity, displacement, etc? Yes! Consider the integral: a(x) dx It s not immediately obvious what it s physical meaning is, but let s try to investigate...

17 1-D Kinematics with varying acceleration Using the chain rule: a dx = a dx = dv dt dx dv dx dx dt dx

18 1-D Kinematics with varying acceleration Using the chain rule: a dx = a dx = dv dt dx dv dx dx dt dx Rearranging: a dx = Chain rule again, and using v = dx ( ) dx dv dt dx dx dt : a dx = v dv

19 1-D Kinematics with varying acceleration xf x i a dx = vf v dv = v 2 v i 2 v f v i o, we find that a(x) dx does mean something physically: IF a is a constant: 1 2 (v f 2 vi 2 ) = a dx 1 2 (v f 2 vi 2 ) = a x vf 2 = vi 2 + 2a x. If a varies with x, all we need to do is evaluate the integral a(x) dx.

20 1-D Kinematics with varying acceleration An asteroid falls in a straight line toward the un, starting from rest when it is 1.00 million km from the un. Its acceleration is given by, a = k where x is the distance from the un to the x 2 asteroid, and k = m 3 /s 2 is a constant. After it has fallen through half-a-million km, what is its speed?

21 Math you will need for 2-Dimensions Before going into motion in 2 dimensions, we will review some things about vectors.

22 Vectors scalar A scalar quantity indicates an amount. It is represented by a real number. (Assuming it is a physical quantity.) vector A vector quantity indicates both an amount and a direction. It is represented more than one real number. (Assuming it is a physical quantity.) There are many ways to represent a vector. a magnitude and (an) angle(s) magnitudes in several perpendicular directions

23 Representing Vectors: Angles Bearing angles Example, a plane flies 750 km h 1 at a bearing of 70 N E Generic reference angles A baseball is thrown at 10 ms 1 30 above the horizontal.

24 Representing Vectors: Unit Vectors Magnitudes in several perpendicular directions: using unit vectors. Unit vectors have a magnitude of one unit.

25 Representing Vectors: Unit Vectors Magnitudes in several perpendicular directions: using unit vectors. Unit vectors have a magnitude of one unit. A set of perpendicular unit vectors defines a basis or decomposition of a vector space.

26 Representing Vectors: Unit Vectors Magnitudes in several perpendicular directions: using unit vectors. Unit vectors have a magnitude of one unit. A set of perpendicular unit vectors defines a basis or decomposition of a vector space. In two dimensions, a pair of perpendicular unit vectors are usually denoted i and j (or sometimes ˆx, ŷ).

27 Representing Vectors: Unit Vectors Perpendicular unit vectors are usually denoted i and j (or sometimes ˆx, ŷ). A generic 2 dimensional vector can be written as v = ai + bj, where a and b are numbers.

28 m of two other component vectors A x, which is parallel to the x axis, and A y, whic parallel Components to the y axis. From Figure 3.12b, we see that the three vectors form ght triangle and that A 5 A x 1 A y. We shall often refer to the component a vector A, written A Consider the 2 dimensional x and A y (without the boldface notation). The compo vector A = A x i + A y j, where A x and nt A x represents the projection of A along the x axis, and the component A A y are numbers. presents the projection of A along the y axis. These components can be positiv negative. The component A We then say that A x is x is positive if the component vector A the i-component (or x-component) of x points i e positive x direction and is negative if A x points in the negative x direction. and A y is the j-component (or y-component) of A. ilar statement is made for the component A y. y y A A A y Ay O a u Ax Notice that A x = A cos θ and A y = A sin θ. x O b u Ax x

29 negative. The component A x is positive if the component vector A x points i e Components positive x direction vsand Magnitude is negative if A/ Angle Notation x points in the negative x direction. ilar statement is made for the component A y. Notice that A x = A cos θ and A y = A sin θ. y y A A A y Ay O a u Also notice, Ax x O b u Ax x A = A = A x + A y and ( ) θ = tan 1 Ay A x if the angle is given as shown.

30 Why Vectors? Of course, there is no reason to limit this to two dimensions.

31 Why Vectors? Of course, there is no reason to limit this to two dimensions. With three dimensions we introduce another unit vector k = ẑ. And we can have as many dimensions as we need by adding more perpendicular unit vectors.

32 Why Vectors? Of course, there is no reason to limit this to two dimensions. With three dimensions we introduce another unit vector k = ẑ. And we can have as many dimensions as we need by adding more perpendicular unit vectors. Vectors are the right tool for working in higher dimensions. They have a property that correctly reflects what it means for there to be more than one dimension: that each perpendicular direction is independent of the others.

33 Why Vectors? Of course, there is no reason to limit this to two dimensions. With three dimensions we introduce another unit vector k = ẑ. And we can have as many dimensions as we need by adding more perpendicular unit vectors. Vectors are the right tool for working in higher dimensions. They have a property that correctly reflects what it means for there to be more than one dimension: that each perpendicular direction is independent of the others. This makes life much easier: we will be able to solve for motion in the x direction separately from motion in the y direction.

34 stant. If the position vector is known, the velocity of the particle from Equations 4.3 and 4.6, which give Visualizing Motion in 2 Dimensions Imagine v 5 d an r 5air dx hockey puck moving with horizontally constant velocity: dt dt i^ 1 dy dt j^ 5 v x i^ 1 v y j^ (4.7) vectors, y elocity, in ure, s that nsions two ns in ctions. a b y x x Figure across a table at directio in the y puck, th ponent ponent in the p If it experiences a momentary upward (in the diagram) acceleration, it will have a component of velocity upwards. The horizontal motion remains unchanged!

35 Vectors Properties and Operations Equality Vectors Commutative A = Blaw ifof and addition only if the magnitudes and directions A are the same. (Each component is the same.) Addition A + B When two vectors are added, the sum is indepe tion. (This fact may seem trivial, but as you will s important when vectors are multiplied. Procedures cussed in Chapters 7 and 11.) This property, which c construction in Figure 3.8, is known as the commut 1 B 5 B 1 A A R A B B D B C A A D B C Figure 3.6 When vector B is added to vector A, the resultant R is the vector that runs from the tail of Figure 3.7 Geometric construction for summing four vectors. The resultant vector R is by definition

36 Vectors Properties and A 1 B Operations 5 B 1 A Properties of Addition addition construction in Figure 3.8, is known as the commutative law of addition: Draw B, then add A. A (3.5) B ector B is e resultant R is rom the tail of A + B = B + A (commutative) D B C A (A + B) + C = A + (B + C) (associative) Figure 3.7 Geometric construction for summing four vectors. The shows that A 1 B 5 B 1 A or, in Figure 3.8 This construction 3.3 ome Properties of Vectors resultant vector R is by definition other words, that vector addition is the one that completes the polygon. Add B and C commutative. ; Add A and B Figure 3.9 Geo ; then add the then add C tions for verifyin to law of addition. result to A. the result. A ( A B C ) B C D C B B C A ( ) B C B A B R B A B A A C B B Draw A, then add B. A A

37 e bookkeeping ents Vectors separately. Properties and Operations Doing addition: Almost always the right answer is to break each vector into components and sum each component independently. (3.14) y nt vector are (3.15) R y B y R B, we add all the tor and use the mponents with A y A x A B x x tained from its R x Figure 3.16 This geometric

38 Vectors Properties and Operations Doing addition: Almost always the right answer is to break each vector into components and sum each component independently. Example w = 5 m at 36.9 above the horizontal. u = 17 m at 28.1 above the horizontal. w + u =?

39 The negative of the vector A is defined as the vector that when added to A gives zero for the vector sum. That is, A 1 12 A The vectors A and 2 A have the Vectors same Properties magnitude but point in and opposite Operations directions. Negation ubtracting Vectors The operation of vector subtraction makes use of the definition of the negative of a vector. We define the operation A 2 B as vector 2 B added to vector A : If u = v then u has the same magnitude as v but points in the opposite direction. Figure 3.10a. ubtraction A A B = A + ( B) A 2 B 5 A 1 12 B 2 (3.7) The geometric construction for subtracting two vectors in this way is illustrated in Another way of looking at vector subtraction is to notice that the difference 2 B between two vectors A and B is what you have to add to the second vector would draw We B here if we were adding it to A. A B Vector C A B is the vector we must add to B to obtain A. A B a B Adding B to A is equivalent to subtracting B from A. b B A C A B Figure 3.10 vector B fro tor 2 B is eq vector B an site directio looking at ve

40 ummary free fall varying acceleration vectors vs scalars representing vectors some vector operations: addition, subtraction Collected Homework! due Friday, Oct 6. (Uncollected) Homework erway & Jewett, NEW: Ch 2, onward from page 49. Probs: 53, 57, 59 Ch 3, onward from page 71. Objective Qs 1 & 3; Probs: 31, 55, 65