Nuffield Free-Standing Mathematics Activity Vectors
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Transcript of Nuffield Free-Standing Mathematics Activity Vectors
© Nuffield Foundation 2011
Nuffield Free-Standing Mathematics Activity
Vectors
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
Examples mass distance speed temperature
Vectors have magnitude and direction
Examples displacement velocity acceleration
force momentum
Unit vectorsSuppose 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 = 12i + 5j
i represents a unit vector to the east
and j represents a unit vector to the north
v =
512Column vector notation
Magnitude and direction of a vector
v
v1
v2v =
2
1vv
Magnitude v = 22
21 vv
Direction tan = 1
2vv
=
1
2vv
tan–1
Think about How can you use the triangle to find the magnitude and direction of v?
v =
512
Example
Speed v = 22 512
Direction tan = 125
v
12
5
= 13
= 60.41
= 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 vectorsAdd or subtract the components
ExampleForces acting on an object
(in newtons)
34-
57
where i is a horizontal unit vector to the right
and j is a vertical unit vector upwards
Total force acting on the object
34-
57
83
Think about how to find the total force
To multiply a vector by a scalarMultiply each component by the scalar
ExampleDisplacement s = (in metres)
2-1
s
3s =
6-3
3s
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 equationstauv Equation 1
t) (21 vus Equation 3
221 tt aus Equation 2
where
u = initial velocity
v = final velocity
a = acceleration
t = time taken
s = displacement
Momentum mv 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 LawF = 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 F1 + F2 + F3
F1
F2
F3
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
2.1-0.5
1.52.4
0.6-2.9
vR=
Speed vR = 22 0.62.9
Direction tan = 2.90.6
= 11.7
= 2.96 ms–1
= 0.2068 … vR
2.9
0.6
bearing
The swimmer will travel at 2.96 ms–1 on bearing 102 (nearest )
N
vS=
1.52.4
vC=
2.1-0.5
(ms–1)
Golf ball
O
u =
1625
i = horizontal unit vector
j = vertical unit vector
a =
9.8-0
Find a the velocity at time tb the velocity when t = 2c the ball’s displacement from O, when t = 2
tauv a t
9.8-0
1625
t9.8-16
25
b When t = 2
29.8-1625
v
c 221 s tt au 2
9.8-0
21
1625 tt
24.9-16
25tt
t
When t = 2 s
224.9-216225
(ms–1)
(m)
3.6-25
(ms–1)
12.450
(m)
Skiera Find the skier’s acceleration.
b Find the speed and directionof the skier 20 seconds later.
u =
68-
F =
15-24
a) F = ma
= 60a
15-24
a =
0.25-0.4
tauv b)
200.25-0.4
68-
v
10v
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
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.
Ship
u =
1-2.5R =
12005600-
500300
100x
a Ship travels at a constant velocity u ms–1
This means there is no acceleration
Ship
u =
1-2.5R =
12005600-
F =
1200-5600
200
21
1-2.5 tt
tt
-2.5
t
t - 500
2.5300
500300r
tt
-2.5
t = 400
x = 300 + 2.5t = 300 + 2.5 400 = 1300
500 – t = 100
O
500300
tt
-2.5
r
100x
Ship’s initial position vector r
500300
221 tt aus b i Displacement
At time t,
b ii When ship reaches
100x
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?