Components of a Vector

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1 Vectors (Ch. 1) A vector is a quantity that has a magnitude and a direction. Examples: velocity, displacement, force, acceleration, momentum Examples of scalars: speed, temperature, mass, length, time.

2 Displacement

3 Components of a Vector A = A x î + A y ĵ

4 Clicker Question What are the x- and y- components C x and C y of vector A. C x = 3 cm, C y = 1 cm B. C x = 4 cm, C y = 2 cm C. C x = 2 cm, C y = 1 cm D. C x = 3 cm, C y = 1 cm E. C x = 1 cm, C y = 1 cm

5 Clicker Question Q1.1 What are the x and y components of the vector E? A. E x = E cos β, E y = E sin β B. E x = E sin β, E y = E cos β C. E x = E cos β, E y = E sin β D. E x = E sin β, E y = E cos β E. E x = E cos β, E y = E sin β

6 Vector Addition If the two vectors are not parallel (or anti-parallel), you can not simply add (or subtract) the magnitude of the two vectors together. Need to add components together. A + B =(A x + B x )î +(A y + B y )ĵ

7 Clicker Question The vectors vectors is A, B, and C S = A + B + C are shown. The vector sum of these three What is the value of S x, the x- component of S? A) +2 B) -4 A B C C) -2 D) +3 E) None of these

8 Q1.4 Clicker Question Which of the following statements is correct for any two vectors A and B? A. the magnitude of A + B is A + B B. the magnitude of A + B is A B C. the magnitude of A + B is greater than or equal to A B D. the magnitude of A + B is greater than the magnitude of A B E. the magnitude of A + B is A 2 + B 2

9 Chalkboard Question: What is the magnitude and direchon (degrees north of due East) of the overall displacement?

10 Chapter 3: 2D (or 3D) Kinematics Main concepts of the previous chapter carry over, but now we need to worry about vectors. In 2D (Cartesian): r(t) = x(t)î + y(t)ĵ v = dr dt = dx dt î + dy dt ĵ a = dv dt = dv x dt î + dv y dt ĵ a x = dv x dt a y = dv y dt

11 Trajectories and Projectile Motion The 1D kinematics equation can be generalized using vectors: For constant acceleration: x(t) =x 0 + v 0x t a xt 2 y(t) =y 0 + v 0y t a yt 2 r(t) =r 0 + v 0 t at2 Projectile Motion due to Gravity: ~a = gĵ x(t) =x 0 + v 0x t y(t) =y 0 + v 0y t 1 2 gt2

12 Clicker Question

Clicker Question You are standing near the edge of a tall building. You throw one ball horizontally with and simultaneously drop another ball from rest (such that it falls down the building).

13 Clicker Question You are standing near the edge of a tall building. You throw one ball horizontally with and simultaneously drop another ball from rest (such that it falls down the building). Which ball hits the ground first (neglect air resistance and assume the ground is not sloped)? 1. The ball thrown horizontally hits the ground first. 2. The ball dropped from rest hits the ground first. 3. The two balls hit the ground at the same Hme.

14 Trajectory A soccer ball is kicked with an initial speed v 0, with the velocity making an angle with respect to the horizontal direction α. Determine y(x) and the range. Initial Conditions: x 0 = y 0 =0 = v 0 cos = v 0 sin v 0x v 0y x(t) =(v 0 cos )t t = y(t) =(v 0 sin )t 1 2 gt2 x v 0 cos x(t) =x 0 + v 0x t y(t) =y 0 + v 0y t 1 2 gt2 y(x) =x tan g 2v 2 0 cos2 x 2

15 Parabolic Trajectory y(x) =x tan g 2v 2 0 cos2 x 2

16 Range of soccer ball: Use trajectory equation y(x): y(x) =x tan g 2v 2 0 cos2 x 2 Final horizontal position when ball hits ground: y f (x f )=0 x =2v 2 0 cos 2 tan /g =2v 2 0 sin cos /g angle for maximum range: dx f d =0= d d v 2 0 sin(2 ) g cos(2 ) max = 45

17 Clicker Question

18 Chalkboard Question A tennis player serves the ball by hitting it over his head, 2.5 m above the ground, giving the ball a horizontal velocity. The ball has to clear the net which is 15 m away and 0.9 m high. The ball must hit the ground within 7 meters of the net. What is the minimum and maximum initial velocities such that the ball is served properly?

19 Clicker Question In a game of shoot the monkey, a monkey jumps off the cliff just as a person is trying to shoot him with a toy gun. How should the person aim such that the toy bullet hits the monkey? a) aim directly at the monkey b) aim a little above the monkey c) aim a little below the monkey shoot- monkey- sim.jar

20 Chalkboard Question A skier jumps off a ramp such that he is initially moving horizontally with a speed v 0. The incline of the mountain is θ. How far away from the jump does the skier land?

21 Chalkboard Question A person is a distance L=10 m away from the bottom of a cliff of height h=15 m. She wishes to thrown an egg up to the top of the cliff such that the egg lands horizontally on the ground. With what speed and angle (w.r.t the ground) should she throw the egg?

22 Chalkboard Question A person is a distance L=10 m away from the bottom of a cliff of height h=15 m. She wishes to thrown an egg up to the top of the cliff such that the egg lands horizontally on the ground. With what speed and angle (w.r.t the ground) should she throw the egg? Want v 0 and θ: = tan 1 2h L = 71.5 v 0 = 18.1 m/s

23 Clicker Question A car is driving around in a circle of radius R with a constant speed. In which direction is the car s acceleration? 1. The acceleration is zero. 2. The acceleration is directed towards the center of the circle. 3. The acceleration is directed radially away from the center of the circle. 4. None of the above

24 Centripetal (radial) Acceleration For circular motion, the object is always accelerating. v = v a = ~a = d~v dt = v d dt = v 2 T = v2 R a cen = v2 R

25 Clicker Question Q3.11 You drive a race car around a circular track of radius 100 m at a constant speed of 100 km/h. If you then drive the same car around a different circular track of radius 200 m at a constant speed of 200 km/h, your acceleration will be A. 8 times greater B. 4 times greater C. twice as great D. the same E. half as great

26 High-G Training 20 G centrifuge at NASA Ames Research Center (8.7 m radius) Astronauts can handle roughly 6g vertically and up to about 17g horizontally (eyes in) For 17g, v=38 m/s; rpm = 86!

27 Chalkboard Question An object is orbiting the Earth just above the Earth s atmosphere (still near the surface relative to the Earth s radius). Determine its orbital velocity and orbital period. (Radius of Earth is 6,600 km)

28 Relative Velocity (Galilean Velocity Transformation) Consider a bus driving down the road at 20 mph and a person walking along the aisle in the bus (towards the front) with a velocity of 2 mph. How fast is the person moving with respect to the road? v r = 20 mph + 2 mph = 22 mph If you consider the bus as a separate frame of reference moving with a velocity u=20 mph with respect to the frame of reference of the stationary road, then v = u + v, where v is the velocity of the person with respect to the bus frame. In two dimensions, the relationship between velocities becomes ~v = ~u + ~v 0

29 Chalkboard Question A swimmer can swim 3 m/s with respect to still water. If the swimmer is swimming in a river with a current of 2 m/s, how long does it take to swim 12 meters downstream and then back upstream? Does it take longer or shorter than if there were no current? ~v = ~u + ~v 0 u =2m/s v 0 = ±3 m/s t = t down + t up t = 12 m 5m/s + 12 m 1m/s t = 14.4 s

30 Chalkboard Question Now the swimmer wants to swim directly across the same river (of width 12 meters) and back. How long does it take? θ ~v = ~u + ~v 0 v x =0) u = v 0 sin =sin 1 (u/v 0 )=sin 1 (2/3) = 41.8 ) v y = v 0 y = 3 cos(41.8 )=2.24 m/s t = s = 10.7 s

31 ~v = ~u + ~v 0 Consider an airplane flying in a crosswind. If the plane is traveling with respect to the air with a velocity and the air is moving with respect to the ground with a velocity then the velocity of the plane wrt the ground is v P/E = v P/A + v A/E = tan 1 v A/E v P/A = 22.6 ~v 0 = ~v P/A ~u = ~v A/E If the plane is facing north, the angle between its velocity and north is

What angle should the echelon make with respect to the direchon of the road?

32 Cycling Echelon Suppose a group of cyclist are riding at 20 mph (say the y direchon) in a cross wind (perpendicular to their direchon of mohon, say the x direchon) with a wind- speed of 15 mph. What angle should the echelon make with respect to the direchon of the road? In frame where cyclists are at rest: ~v w 0 = ~v w ~u c v 0 w,x = 15 mph v 0 w,y = 20 mph = tan ~v = ~u + ~v 0 = 37

Kinematics in Two-Dimensions

Slide 1 / 92 Slide 2 / 92 Kinematics in Two-Dimensions www.njctl.org Slide 3 / 92 How to Use this File Each topic is composed of brief direct instruction There are formative assessment questions after