Need to have some new mathematical techniques to do this: however you may need to revise your basic trigonometry. Basic Trigonometry

Transcription

1 Kinematics in Two Dimensions Kinematics in 2-dimensions. By the end of this you will 1. Remember your Trigonometry 2. Know how to handle vectors 3. be able to handle problems in 2-dimensions 4. understand projectile motion Swish! by alanfreed Two (or Three) Dimensional Motion For example: Projectiles, Planets, Pendulum,... Need to have some new mathematical techniques to do this: however you may need to revise your basic trigonometry Basic Definitions of trig relationships A useful mnemomic SohCahToa θ = a R Sin = opposite = d hypotenuse R Cos = adjacent = R-δ hypotenuse R Tan = opposite = d adjacent R-δ Basic Trig. Useful relationships sin 2 θ + cos 2 θ = 1 (all values of θ) tan θ = sin θ / cos θ Basic Trigonometry Page 1 of 10

2 These are not so useful cot θ = 1/tan θ sec θ = 1/cos θ cosec θ = 1/sin θ Special Values sin(45 0 ) = sin(π/4) = 1/ 2 cos(45 0 ) = cos(π/4) = 1/ 2 tan(45 0 ) = tan(π/4) = 1 sin(30 0 ) = sin(π/6) = 1/2 cos(60 0 ) = cos(π/3) = 1/2 sin(60 0 ) = cos(30 0 ) = 3/2 tan(30 0 ) = tan(π/3) = 1/ 3 tan(60 0 ) = tan(π/6) = 3 sin(90 0 ) = sin(π/2) = 1 cos(90 0 ) = cos(π/2) = 0 tan(90 0 ) = tan(π/2) = These are worth writing down, but they are very easy to work out. We often have a convenient simplification when the angles are small θ = a/r ~ sin(θ) = d/r ~ tan(θ) = d (R-δ) cos(θ) ~ 1 Note that this only works if we measure angles in radians: 2π radians = Vectors Need new mathematics to describe three dimensional objects: Scalars: quantities with only magnitude Vectors have direction as well Scalars Vectors Temperature Force Page 2 of 10

3 Speed Density Time? Velocity Acceleration Time? Need a new symbol: can use ~ a which is obvious a which gets confused with other symbols a, which is hard to read We will use both ~ a and a, Need to be able to describe vectors in terms of scalar quantities: can do this in terms of components: the projection of the vector along each axis These are the components of the vector ~ a Note that the components of a vector are scalars Also can do this in terms of length of the vector and angle(s). These two descriptions are related a x = a cos(θ) a y = a sin(θ) Page 3 of 10

4 Adding vectors: put them nose to tail. Easy diagramatically If we want to add them algebraically, we just add the components: means ~ c = a~ + ~ b c x = a x + b x c y = a y + b y To subtract vectors, Flip the vector round: the negative of a vector must add to the vector to give zero ~ a - ~ a = ~ a + ( -a) ~ = 0 Note that this means that the negative of a vector just has all its components reversed The length of a vector is given by Pythagoras: 3-D analog of Pythagoras: Page 4 of 10

5 q j~j r = x 2 + y 2 + z 2 Write this as Note: The length of a vector is a scalar. Displacement is a vector. The length of the displacement vector is the distance Examples e.g. a boat sails 10 km North -East and 5 km South: how must it sail to get to its start? How do we write this as a vector sum? Page 5 of 10

6 N.E means "equal components along the x and y directions" so first step is e.g. Motion by a car. A car travels 5 km N, 10 km E, and then 15 km S. The components of vector ~ a that describes this are 1. a x = -10 a y = a x = 10 a y = a x = 10 a y = a x = -10 a y = -10 We write this as (can usually be treated as 2-D) Projectile Motion Page 6 of 10

7 Treat position, r, velocity v and acceleration as 2-D vectors. In general,motion in one direction can be treated independently of motion in a second. Page 7 of 10

8 We can also see this quantitatively In general there are three independent vector quantities: The position r, the velocity v and the acceleration a. However we have to treat the components separately. It is easiest to treat the two motions as independent 1-D motions vy = v vx = v 0x + a x t 0y + a y t 1 1 x = v 0x t + 2 a x t 2 y = v 0y t + a t 2 2 y ay = Àg(usually) a x = 0(usually) À2 = À9:8ms Page 8 of 10

9 À Can combine these to give (e.g) an equation for the range R: What is value of t when the height y = 0? t = 2v 0 sin ( Ò) g (or t = 0: what does this mean?) Hence Range R = v 2 0 sin ( 2Ò) g e.g. a ball is thrown at 15 ms -1, at 30 0 to the horizontal. How far does it go How long is it in the air? Relative Velocity Example: a woman who can swim at 2 m/s is swimming How fast does she swim upstream? How fast does she swim downstream? How about a round trip? in a river which flows at.7 m/s. A woman swims 100 m upstream at 2 m/s in a river with a current of.7 m/s, and then 100 m downstream to return to her starting point. Compared to swimming 200 m in still water, does her journey 1. Take longer? 2. Take a shorter time? 3. Take exactly the same time? As a somewhat more sophisticated example: a woman who can Page 9 of 10

10 woman who can swim at 2 m/s is swimming in a river which flows at.7 m/s. At what angle should she swim to reach a point on the opposite bank immediately opposite the point from which she starts? Now we want to describe why things move, not just how. Page 10 of 10