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 distance speed temperature Examples mass Vectorshave magnitude anddirection displacement velocity acceleration Examples force momentum
v = 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 isvms–1 wherev= 12i+ 5j irepresents a unit vectorto the east and jrepresents a unit vectorto the north Column vector notation
v v= tan–1 Magnitude and direction of a vector v = v2 v1 Think about How can you use the triangle to find the magnitude and direction of v? Magnitude tan = Direction =
N bearing v 5 12 = v= Example Think about How can you find the speed and direction of the yacht? v= = 13 Speed tan = Direction = 22.6 The yacht is sailing at 13 ms–1 on bearing 067 (nearest )
To add or subtract vectors Add or subtract the components Example Forces acting on an object (in newtons) wherei is a horizontal unit vector to the right andjis a vertical unit vector upwards Think about how to find the total force Total force acting on the object
Example Displacements = (in metres) s 3s To multiply a vector by a scalar Multiply each component by the scalar Think about What do you get if you multiply both components of the vector by 3? 3s = Multiplying by 3 gives a displacement 3 times as big in the same direction
Equation 1 where u = initial velocity v= final velocity a = acceleration Equation 2 t= time taken Equation 3 s = displacement Constant acceleration equations mv Momentum is a vector
Forces and acceleration F1 Resultant force is the sum of the forces acting on a body, in this case F1 + F2 + F3 F2 F3 Newton’s First Law 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 Resultant forcecauses acceleration Newton’s Third Law Action and reaction are equal and opposite. This means if a body A exerts a force on a body B, then B exerts an equal and opposite force on A.
vC= vR= Speed N bearing 2.9 tan = 0.6 vR vS= Swimmer Find the magnitude and direction of the swimmer’s resultant velocity. Resultant velocity vR= (ms–1) i= unit vectorto the eastj= unit vectorto the north = 2.96 ms–1 = 0.2068 … Direction = 11.7 The swimmer will travel at 2.96 ms–1 on bearing 102 (nearest )
O a u= a = v (ms–1) c s (m) Golf ball Find a the velocity at time t b the velocity when t= 2 c the ball’s displacement from O, when t = 2 i= horizontal unit vectorj= vertical unit vector (ms–1) b When t = 2 (m) When t = 2
b) = 60a u = F= a = Skier a Find the skier’s acceleration. b Find the speed and direction of the skier 20 seconds later. 60 kg i= unit vectorto the eastj= unit vectorto the north a)F = ma (ms–2) (ms–1) The skier is travelling at 1 ms–1 to the north.
u = R = Ship Ship travels at a constant velocity u ms–1 a What is the force, F, from the tug? b Ship’s initial position vector r i Find the position vector of the boat at time t. ii The ship is aiming for a buoy which has position vector Assuming the ship reaches the buoy, find x.
u = R = F = r O Ship a Ship travels at a constant velocity u ms–1 This means there is no acceleration b i Displacement Ship’s initial position vector r At time t, When ship reaches b ii 500 – t = 100 t = 400 x = 300 + 2.5t = 300 + 2.5 400 = 1300
Reflect on your work • How have you used the fact that i and jare 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?