#MEIConf2018. Variable Acceleration. Sharon Tripconey

Size: px

Start display at page:

Download "#MEIConf2018. Variable Acceleration. Sharon Tripconey"

Blaise Stone
5 years ago
Views:

1 @MEIConference #MEIConf2018

2 #MEIConf2018 Variable Acceleration Sharon Tripconey

3 #MEIConf2018 Kinematics in A level Maths Ref Q1 Q2 Q3 Q4 Q5 Content description [Understand and use the language of kinematics: position; displacement; distance travelled; velocity; speed; acceleration] [Understand, use and interpret graphs in kinematics for motion in a straight line: displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph] [Understand, use and derive the formulae for constant acceleration for motion in a straight line]; extend to 2 dimensions using vectors [Use calculus in kinematics for motion in a straight line: vv = dddd dddd, aa = dddd = dd22 rr dddd ddtt 22, rr = vv dddd, vv = aa dddd ]; extend to 2 dimensions using vectors Model motion under gravity in a vertical plane using vectors; projectiles

4 #MEIConf2018 Displacement, distance & distance travelled displacement 0 time

5 Velocity and speed #MEIConf2018 velocity 0 time

6 #MEIConf2018 Displacement, distance & distance What would the position graph look like? 0 travelled time distance displacement distance travelled What is significant about the gradient in each of these graphs?

7 Velocity and speed #MEIConf2018 velocity speed 0 time

8 Extending basic ideas #MEIConf2018 The instantaneous velocity is the gradient of the displacement time graph displacement s m time t s

9 Motion in a straight line #MEIConf2018

10 #MEIConf2018 Moving man

11 Motion graphs activity #MEIConf2018

12 #MEIConf2018 Motion graphs summary Motion Graph Gradient Area Notes Velocity Not significant Velocity-time Acceleration Displacement Displacementtime Accelerationtime Rate of change of acceleration Velocity Vertical axis can be positive or negative Areas below the time axis represent negative displacement. v=0 indicates a possible change in direction

13 #MEIConf2018 Calculus displacement x x = vdt velocity v = dx dt v = adt acceleration dv a = = dt d 2 dt x 2 a

14 #MEIConf2018 Making links: constant acceleration 2 d s dt s = ut + at 2 ds v = u + at dt dv = a (constant) dt dv a dt??? a = v ds

15 #MEIConf2018 Extending to 2 dimensions using vectors In your handout there are four questions. Which looks hardest? Rank the questions in order of perceived difficulty Now do the questions and rank them in order of actual difficulty Did your lists agree?

16 #MEIConf A force FF acts on a particle of mass 2 kg. Given that FF = 4tti + 6j and the particle has an initial velocity of 5j at the origin, find the velocity and displacement when tt = 3. F= ma 4ti+ 6j= 2a a= 2ti+ 3j v= 2ti+ 3jdt 2 = + + i 3tj c When t = 0, v = 5 j: 5= j c t v t i+ t+ j 2 = (3 5) At t = 3, v= 9i+ 14j s= vdt 3 2 t 3t = i+ ( + 5) t j+ c 3 2 When t = 0, s = 0 : 3 2 t 3t s= i+ ( + 5) t j At t = 3, s= 9i+ j 2

17 #MEIConf Particle PP is moving in the xxxx plane with the origin at OO. The position vector of PP with respect to OO is r = 2tt 3 i + 3tt 2 j. Find the velocity vector and the acceleration vector for PP and hence the magnitudes of the velocity and acceleration when tt = r = t i+ t j v = 2 3 dr dt = t i tj dv a = dt = 12ti+ 6j At t = 3, v= 54i+ 18j v = = 56.9ms 2 2 a= 36i+ 6j a 1 = = ms 2

18 3. The position vector of a radio controlled car is r = (2tt 1)i tt 2 j The motion is initially in the positive horizontal direction #MEIConf2018 a) Find the velocity at time tt. b) Find the initial direction of the motion. c) Show that the acceleration is constant. d) Explain why the car can never move in a direction that is perpendicular to the original direction. 2 a) r = (2t 1) i t j dv c) a = dr dt v = dt = 2j = 2i 2tj b) At t = 0, v= 2i d) The initial velocity is horizontal, so the motion would need to become vertical (i.e. no horizontal component). However, there is no horizontal acceleration, so the velocity will always have a horizontal component of 2i.

19 GeoGebra: Question 3 #MEIConf2018

20 4. A particle moves such that at time tt seconds its acceleration is given by (2cccccc tt i 5ssssss ttj) ms -1. #MEIConf2018 a) The mass of the particle is 6 kg. Find the magnitude of the resultant force on the particle when tt = 0. b) When tt = 0, the velocity of the particle is (2i + 10j) ms -1. Find an expression for the velocity of the particle at time tt. a) F= ma At t = 0, = 6(2 costi 5sin tj) = 12 costi 30sin tj) F= 12i Therefore the magnitude of the resultant force is 12 N. b) v= adt = 2sinti+ 5costj+ c When t = 0, v=2 i+10j 2 i+10j= 5j+ c c= 2 i+5j v= 2sinti+ 5costj+ 2 i+5j = 2(sin t+ 1) i+ 5(cos t+ 1) j

21 Sustained PD Courses #MEIConf2018 AMSP sustained PD courses blend online learning with face-to-face study days and the use of dedicated resources. Teaching A Level Mathematics Teaching Further Mathematics Teaching Mechanics Teaching Statistics Teaching Discrete Mathematics TAM TFM1 & TFM2 TM1 & TM2 TS1 & TS2 TD For full details see:

22 #MEIConf2018 About MEI Registered charity committed to improving mathematics education Independent UK curriculum development body We offer continuing professional development courses, provide specialist tuition for students and work with employers to enhance mathematical skills in the workplace We also pioneer the development of innovative teaching and learning resources

23 Variable acceleration Sharon Tripconey

24 displacement 0 time velocity 0 time

26 Variable Acceleration Questions Question 1 A force F acts on a particle of mass 2 kg. Given that F = 4ti + 6j and the particle has an initial velocity of 5j at the origin, find the velocity and displacement when t = 3. Question 2 Particle P is moving in the xy plane with the origin at O. The position vector of P with respect to O is r = 2t 3 i + 3t 2 j. Find the velocity vector and the acceleration vector for P and hence the magnitudes of the velocity and acceleration when t = 3. Question 3 The position vector of a radio controlled car is r = (2t 1)i t 2 j a) Find the velocity at time t. b) Find the initial direction of the motion. c) Show that the acceleration is constant. d) Explain why the car can never move in a direction that is perpendicular to the original direction. Question 4 A particle moves such that at time t seconds its acceleration is given by (2cos ti 5sin tj) ms -2 a) The mass of the particle is 6 kg. Find the magnitude of the resultant force on the particle when t = 0. b) When t = 0, the velocity of the particle is (2i + 10j) ms -1. Find an expression for the velocity of the particle at time t. Teaching Mechanics MEI 2018

27 MEI How to Guides for GeoGebra Mechanics: Creating a position/velocity diagram in 2D in GeoGebra Problem: A boat P has position vector (2i 8j) km when t = 0 and is moving with constant velocity ( 4i+8j) km h -1. (i) At time t hours the position vector p. Write down p in terms of t. Calculate the speed of P. A second boat Q has position vector q = 18i + 12j t (6i + 8j). (ii) Find the value of t when P is due west of Q and the distance between P and Q when P is due west of Q. Adding a slider for the time and displaying the particle 1 2 Add a slider for the time (10 th menu). Set the name to t and set the min to 0. Using the input bar enter: P=(2-4t,-8+8t). Creating the velocity vector (from the origin) and translating it to the particle Using the input bar enter: v=(-4,8) Select Vector from Point (3 rd menu) then select P and v. Hide the vector v (at the origin) and the point P. Right-click on P and select Trace On Using the input bar enter: Speed=Length[v] Displaying the second particle and finding the distance 9 10 Using the input bar enter: Q=(18-6t,12-8t) Use Segment between Two Points to create the line segment PQ. View on GeoGebraTube: 1 of 1 MEI 24/10/2014 TB v1.1

28 Resources, ideas and support Moving Man Thinking constantly MEI Casio tasks for Mechanics (Tasks 4, 5 & 6) FMSP PD videos: Mechanics (Exploring kinematic graphs) Motion graphs cards activity Teaching Mechanics (AMSP Sustained PD course)

MEI Conference Preparing to teach motion graphs. Sharon Tripconey.

MEI Conference Preparing to teach motion graphs. Sharon Tripconey. MEI Conference 2016 Preparing to teach motion graphs Sharon Tripconey sharon.tripconey@mei.org.uk 1 of 5 Preparing to teach motion graphs June 2016 MEI Kinematics content extracts for reformed Mathematics