Introduction to Mechanics Motion in 2 Dimensions

Transcription

1 Introduction to Mechanics Motion in 2 Dimensions Lana heridan De Anza College Jan 31, 2018

2 Last time vectors and trig

3 Overview introduction to motion in 2 dimensions constant velocity in 2 dimensions relative motion

4 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = b (A) yes (B) no

5 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = b (A) yes (B) no

6 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = n (A) yes (B) no

7 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = n (A) yes (B) no

8 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = n (A) yes (B) no

9 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = n (A) yes (B) no

10 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = b + n (A) yes (B) no

11 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = b + n (A) yes (B) no

12 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = n b (A) yes (B) no

13 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibly be a valid equation? a = n b (A) yes (B) no

14 Motion in 2 Dimensions o far we have looked at motion in 1 dimension only, motion along a straight line. However, motion on a plane (2 dimensions), or through space (3 dimensions) obeys the same equations. We will now focus on 2 dimensional motion.

15 from Equations 4.3 and 4.6, which give Motion in 2 directions Imagine v 5 d an r 5air dx hockey puck moving with horizontally constant dt dt i^ 1 dy dt j^ 5 v x i^ 1 v y j^ (4.7) velocity: 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! 1 Figure from erway & Jewett, 9th ed.

16 Direction and Motion When we say something is moving, we mean that it is moving relative to something else. In order to describe measurements of where something is how fast it is moving we must have reference frames. In 2 dimensions we need to choose a pair of perpendicular directions to be our x and y axes.

17 nts Motion of Vectors in 2 directions: Components of velocity can be resolved into horizontal and Motion in perpendicular directions can be analyzed separately. ents. A vertical force (gravity) does not affect horizontal motion. The horizontal component of velocity is constant. 1 Drawing by Paul Hewitt, via Pearson.

18 Constant Velocity in 2 Dimensions and Consider a turtle that moves with a constant velocity. as the x and y equation y d = v 0 t y = d sin θ O θ = 25 x = d cos θ x We can find the distance it travels (a) by using the equation d = v 0 t. FIGURE 4 1 Constant velocity A turtle walks from the origin with a speed of v 0 = 0.26 m/s. (a) In a 1 Figure thusfrom the xwalker, and y displacements Physics. are x = d cos u, y = d sin u. (b) Equiv

19 Constant Velocity in 2 Dimensions and However, if know how far the turtle has traveled in the x direction and y direction separately, this gives us the components of the the turtle s displacement vector, r. as the x and y equations of m y d = v 0 t y = d sin θ O θ = 25 x = d cos θ x v 0y = v We can find the distance it travels (a) in the x-direction using trigonometry: x = d cos θ. FIGURE 4 1 Constant velocity A turtle walks from the origin with a speed of v 0 = 0.26 m/s. (a) In a time t th thus the x and y displacements are x = d cos u, y = d sin u. (b) Equivalently, and v 0y = v 0 sin u; hence x = v 0x t and y = v 0y t. And in the y-direction: y = d sin θ. 1 Figure from Walker, Physics.

20 Constant Velocity in 2 Dimensions x and y equations of motion. y = y 0 + v 0y t y 4 2 d sin θ y = v 0y t v 0y = v 0 sin θ O v 0 θ = 25 v 0x = v 0 cos θ x = v 0x t x Or, we can find the distance it travels in the x-direction by considering what is its rate of change of x-position with time! 0.26 m/s. (a) In a time t the turtle moves through a straight-line distance of d = v 0 t; = d sin u. (b) Equivalently, the turtle s x and y components of velocity are v 0x = v 0 cos u (b) v 0x = x t = v 0 cos θ x = (v 0 cos θ)t And in the y-direction: v 0y = y t = v 0 sin θ y = (v 0 sin θ)t 1 Figure from Walker, Physics.

21 Relative Motion We can use the notion of motion in 2 dimensions to consider how one object moves relative to something else. All motion is relative. Our reference frame tells us what is a fixed position. An example of a reference for time and space might be picking an object, declaring that it is at rest, and describing the motion of all objects relative to that.

22 Intuitive Example for Relative Velocities tors ocity relative to ds on the relative to the d s velocity. 1 Figure by Paul Hewitt.

23 Intuitive Example Now, imagine an airplane that is flying North at 80 km/h but is blown off course by a cross wind going East at 60 km/h. How fast is the airplane moving? In which direction? ketch: 1 Figure by Paul Hewitt.

24 Intuitive Example Now, imagine an airplane that is flying North at 80 km/h but is blown off course by a cross wind going East at 60 km/h. How fast is the airplane moving? In which direction? ketch: 1 Figure by Paul Hewitt.

25 Intuitive Example Now, imagine an airplane that is flying North at 80 km/h but is blown off course by a cross wind going East at 60 km/h. How fast is the airplane moving? In which direction? ketch: Hypothesis: It will travel to the North-East, at a speed greater than 80 km/h, but less than = 140 km/h. 1 Figure by Paul Hewitt.

26 Intuitive Example

27 Intuitive Example trategy: vector addition! In this case, the two vectors are at right-angles. We can use the Pythagorean theorem.

28 5 Projectile Motion Intuitive Example 5.2 Velocity Vectors An 80-km/h airplane flying in a 60-km/h crosswind has a resultant speed of 100 km/h relative to the ground. trategy: vector addition! In this case, the two vectors are at right-angles. We can use the Pythagorean theorem. v = 100 km/h at 36.9 East of North (or 53.1 North of East)

29 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h relative to the water. The frame water B, in which themoves river to the has right a uniform speed of with a constant velocity v BA. The 5.00 km/h due east relative vector to rthe PA is the Earth. particle s If position the boat heads due vector relative to north, determine the velocity of the boat A, and relative r P B is its to an observer standing on either bank. 1 position vector relative to B. ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r ketch: peed of ver has a e Earth. locity of bank. s a river will not end up W N vbr E u vre vbe W N E vbr v re u vbe you rela- a 2 Page 97, erway & Jewett b

30 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h relative to the water. The frame water B, in which themoves river to the has right a uniform speed of with a constant velocity v BA. The 5.00 km/h due east relative vector to rthe PA is the Earth. particle s If position the boat heads due vector relative to north, determine the velocity of the boat A, and relative r P B is its to an observer standing on either bank. 1 position vector relative to B. v br = 10.0 km/h ketch: v re = 5.00 km/h ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe 2 Page 97, erway & Jewett W b N E vbr v re u vbe

31 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h relative to the water. The frame water B, in which themoves river to the has right a uniform speed of with a constant velocity v BA. The 5.00 km/h due east relative vector to rthe PA is the Earth. particle s If position the boat heads due vector relative to north, determine the velocity of the boat A, and relative r P B is its to an observer standing on either bank. 1 position vector relative to B. v br = 10.0 km/h ketch: v re = 5.00 km/h ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe 2 Page 97, erway & Jewett W b N imply use vector addition to find v re v be. E vbr u vbe v be = = 11.2 km/h

32 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h relative to the water. The frame water B, in which themoves river to the has right a uniform speed of with a constant velocity v BA. The 5.00 km/h due east relative vector to rthe PA is the Earth. particle s If position the boat heads due vector relative to north, determine the velocity of the boat A, and relative r P B is its to an observer standing on either bank. 1 position vector relative to B. v br = 10.0 km/h ketch: v re = 5.00 km/h ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe 2 Page 97, erway & Jewett W b N imply use vector addition to find v re v be. E vbr u vbe v be = = 11.2 km/h ( ) 5 θ = tan 1 =

33 Relative Motion Practice: Concepts Three motorboats are crossing a river. All have the same speed relative to the water. (a) ketch resultant vectors for each boat showing the speed and direction of the boats. Rank them from most to least for (b) the fastest ride. (c) the time to reach the opposite shore. 1 Figure and question from Hewitt, Conceptual Physics.

34 ummary vectors motion in 2-dimensions motion with constant velocity relative motion Test tomorrow. Homework study for test