© Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Vectors

Is the water skier moving in the same direction as the rope? What forces are acting on the water skier? Which directions are the forces acting in?

Vectors Scalar quantities have magnitude but no direction Examplesmass distancespeedtemperature Vectors have magnitude and direction Examples displacementvelocityacceleration forcemomentum

Unit vectors Suppose the velocity of a yacht has an easterly component of 12 ms –1 and a northerly component of 5 ms –1 The velocity is v ms –1 where v = 12 i + 5 j i represents a unit vector to the east and j represents a unit vector to the north v = Column vector notation

Magnitude and direction of a vector v v1v1 v2v2 v = Magnitude v =v = Direction tan  =  = = tan –1  Think about How can you use the triangle to find the magnitude and direction of v ?

v =v = Example Speed v =v = Direction tan  =  v 12 5  = 13 = = 22.6  bearing The yacht is sailing at 13 ms –1 on bearing 067  (nearest  ) N Think about How can you find the speed and direction of the yacht?

To add or subtract vectors Add or subtract the components Example Forces acting on an object (in newtons) where i is a horizontal unit vector to the right and j is a vertical unit vector upwards Total force acting on the object Think about how to find the total force

To multiply a vector by a scalar Multiply each component by the scalar Example Displacement s = (in metres) s 3 s = 3s3s Multiplying by 3 gives a displacement 3 times as big in the same direction Think about What do you get if you multiply both components of the vector by 3?

Constant acceleration equations Equation 1 Equation 3 Equation 2 where u = initial velocity v = final velocity a = acceleration t = time taken s = displacement Momentum mvmv is a vector

Forces and acceleration Newton’s First Law Resultant force causes acceleration Action and reaction are equal and opposite. A particle will remain at rest or continue to move uniformly in a straight line unless acted upon by a non-zero resultant force. Newton’s Second Law F = ma Newton’s Third Law This means if a body A exerts a force on a body B, then B exerts an equal and opposite force on A. Resultant force is the sum of the forces acting on a body, in this case F 1 + F 2 + F 3 F1F1 F2F2 F3F3

Swimmer i = unit vector to the east j = unit vector to the north Find the magnitude and direction of the swimmer’s resultant velocity. Resultant velocity vR=vR= Speed vR =vR = Direction tan  =  = 11.7  = 2.96 ms –1 = … vRvR  bearing The swimmer will travel at 2.96 ms –1 on bearing 102  (nearest  ) N vS=vS= vC=vC= (ms –1 )

Golf ball O u =u = i = horizontal unit vector j = vertical unit vector a = Find a the velocity at time t b the velocity when t = 2 c the ball’s displacement from O, when t = 2 a b When t = 2 v c When t = 2 s (ms –1 ) (m) (ms –1 ) (m)

Skier aFind the skier’s acceleration. bFind the speed and direction of the skier 20 seconds later. u = F =F = a) F = ma = 60 a a = b) The skier is travelling at 1 ms –1 to the north. (ms –2 ) (ms –1 ) i = unit vector to the east j = unit vector to the north 60 kg

Ship travels at a constant velocity u ms –1 aWhat is the force, F, from the tug? bShip’s initial position vector r iFind the position vector of the boat at time t. iiThe ship is aiming for a buoy which has position vector Assuming the ship reaches the buoy, find x. Ship u = R =

aShip travels at a constant velocity u ms –1 This means there is no acceleration Ship u = R = F = t = 400 x = t =  400 = – t = 100 O r Ship’s initial position vector r b i Displacement At time t, b ii When ship reaches

Reflect on your work How have you used the fact that i and j are perpendicular unit vectors? Are there any similarities between the problems or the techniques you have used? Can you think of other scenarios which could be tackled using vectors in component form?