Caldervale High School
CfE Higher Physics
Our Dynamic Universe
Summary Notes (Section 1)
Section 1: Motion — Equations and Graphs
Vector and Scalar Definitions
Scalar:
a physical quantity that can be defined by __________________ only.
Vector:
a physical quantity that can be defined by __________________ and
___________________ .
Examples of Scalar Quantities
__________________________________________________________________
Examples of Vector Quantities
_____________________________________________
Distance and Displacement
A walker takes the following path (dotted line).
The dotted line represents the
total _______________
travelled.
The arrow represents the total _____________________ as a straight line from
start to finish with ____________________________ (x) and
______________________ (), East of North.
NOTE:
It is common to represent the direction as a 3 figure bearing.
Page 1
Caldervale High School
CfE Higher Physics
Speed and Velocity
Speed is a ____________ quantity.
Equation …
speed =
Velocity is a ____________ quantity where the direction of the average velocity is
the same as the direction of the overall displacement.
Equation …
velocity =
Worked Example 1
A car moves 30 m, North, then 50 m, West, then 30 m, South in 1 minute.
a) Determine the average speed of the car.
b) Determine the average velocity of the car.
Page 2
Caldervale High School
CfE Higher Physics
Worked Example 2
Use a scale diagram to calculate the displacement of a man who walks 40m, North
then 100m at 120o.
Worked Example 3
Use Pythagoras and trigonometry to calculate the displacement of a man who walks
40m, North then 50m, East then 10m, South.
Page 3
Caldervale High School
CfE Higher Physics
Addition of Vector Quantities
Worked Example
A pilot has selected a course of 100 kmh-1, due West. A wind blowing is blowing
with a velocity of 40 kmh-1, due South. Determine the resultant velocity of the plane.
By Scale Diagram Method
By Pythagoras / Trig Method
Page 4
Caldervale High School
CfE Higher Physics
Resolving Vectors
Components of a Vector
Any single vector can be split into 2 vectors at right angles to each other.
Eg)
Horizontal and Vertical Components
If these 2 component vectors are in line with the horizontal and vertical planes, we
call them “__________________” and “______________________” components.
If we know the size of vector, x, and an angle, , then we can calculate the horizontal
& vertical components from soh or cah.
*In general*
adj component =
opp component =
Page 5
Caldervale High School
CfE Higher Physics
Worked Examples
Calculate the horizontal and vertical components of the following vector quantities:
Objects on a Slope
If an object ________________ in some direction, then the ______________
making it accelerate must be in the same direction. So we often have to calculate the
component of force in the direction of motion.
For example, a mass (m) on a frictionless slope which makes an angle, , with the
horizontal …
The force acting on the mass is due to _______________ giving the downwards
______________ of the mass …
W =
Page 6
Caldervale High School
CfE Higher Physics
BUT the unbalanced force causing the downslope acceleration is the
______________________ __________________________ of ___________ .
So …
Downslope component of weight = opp = hyp sin
Downslope component of weight
=
(perpendicular component of weight = m g cos)
Worked Example
A 2 kg mass is released on a slope which makes a 30o angle with the ground. A
frictional force of 4N acts upslope. Calculate the acceleration of the mass.
Page 7
Caldervale High School
CfE Higher Physics
Sign Convention
In Physics we use positive (+ve) and negative (-ve) to define direction of a vector
quantity.
Let’s consider …
A ball rolls towards a wall with a speed of 5 ms-1. It bounces from the wall and rolls at
4.5 ms-1, in the opposite direction.
So, we say…
If it’s velocity towards the wall = 5 ms-1
then it’s velocity from the wall = _________ .
THINK ABOUT THIS…
Does a positive acceleration always mean speeding up?
… ______________ .
A positive acceleration always means the _________________ is increasing.
THINK ABOUT THIS…
A satellite is orbiting with a steady speed.
Is it accelerating?
Acceleration due to gravity is ALWAYS =
It is up to us to decide in each problem whether this is ________ or ________.
Page 8
Caldervale High School
CfE Higher Physics
Sometimes in problems we may choose the downwards direction as –ve, other times
+ve.
We must
•
state the __________ ____________________ at the start of the problem.
•
stick to the _______ _______ _________________ throughout the problem.
•
A –ve acceleration must not be taken as _______________ __________ .
Worked Example 1
A ball is thrown vertically upwards with a velocity of 5 ms-1. How long will it take to
reach it’s highest point?
1st Choose Sign Convention …
2nd– List & Calculation
Worked Example 2
A ball is pushed up a slope with an initial velocity of 2 ms-1. It decelerates at 0.4 ms-2.
What’s its velocity after 3s, 7s and 10s?
1st Choose Sign Convention
…
(t = 3 s)
(t = 7 s)
(t = 10 s)
Page 9
Caldervale High School
CfE Higher Physics
Motion Graphs
Graphs of Displacement, Velocity and Acceleration
1. Constant Displacement (i.e. object remains stationary)
THINK GRADIENT !!!
Velocity graphs = gradient of ___________________ graphs.
Acceleration graphs = gradient of ____________________ graphs.
2. Constant Velocity
3. Constant +ve Acceleration
4. Constant – ve Acceleration
Page 10
Caldervale High School
CfE Higher Physics
Worked Example 1
Draw the corresponding “acceleration – time” graph from the following “velocity –
time” graph.
Page 11
Caldervale High School
CfE Higher Physics
Worked Example 2
Draw the corresponding “velocity – time” graph from the following “acceleration –
time” graph for an object starting from rest.
Page 12
Caldervale High School
CfE Higher Physics
Worked Example 3 – Object Thrown into Air
Sketch graphs of speed, velocity and acceleration against time for a ball thrown
vertically upwards at 20ms-1 and taking 4 s to land. (Assume “a = 10 ms-2” here.)
Speed
Velocity
Acceleration
a)
What distance does the ball travel during the 4 s?
b)
What is the displacement of the ball after 4 s?
c)
What is the speed of the ball at 4 s?
d)
What is the velocity of the ball at 4s?
Distance travelled = area below a speed-time graph
Displacement = Area in a velocity-time graph
(with areas below the time axis being negative.)
Worked Example 4 – Bouncing Ball
Sketch the velocity-time graph of a ball that is released from a height and bounces
twice. Describe the motion at each stage. (Numerical axes are not required and
assume no energy is lost due to negligible contact time with ground).
Page 13
Caldervale High School
CfE Higher Physics
Equations of Motion
Deriving the Equations of Motion
The following 3 equations can be used for any situation that involves a constant
acceleration (horizontally OR vertically).
GRAPH FOR CONSTANT ACCELERATION (in general)
-1
u = initial velocity (ms )
-1
v = final velocity (ms )
-2
a = acceleration (ms )
s = displacement (m)
t = time (s)
1st Equation:
“a”
=
rate of change of velocity
a
=
(v - u) / t
so
2nd Equation:
3rd Equation:
2
v
2
v
2
v
2
v
=
“s”
=
s
=
s
=
s
=
s
=
area under graph line
SQUARE EQUATION 1
(u + at)
2
=
=
SO …
2
v
=
=
Page 14
Caldervale High School
CfE Higher Physics
Applying Equations of Motion
Worked Example
A helicopter travelling upwards at 2 ms-1 releases a box from a height of 30 m.
Calculate the landing velocity of the box.
Page 15
Caldervale High School
CfE Higher Physics
Projectiles
A “projectile” is an object which is projected (thrown, dropped, fired) into the air or
space.
e.g. marble pushed off the edge of a bench.
e.g. Cannon ball being fired into the air.
These objects travel in a curved path!
A projectile moves horizontally and vertically simultaneously so, for calculations, we
need to separate these motions.
Horizontally: _____________________________________________________
Vertically:
_____________________________________________________
Page 16
Caldervale High School
CfE Higher Physics
Worked Example 1
A ball is projected horizontally off the end of a bench. It hits the floor 3 m from the
base of the bench which is 1.25m high. Calculate
a)
b)
c)
d)
e)
the time of flight.
the velocity of projection.
the horizontal component of velocity as the ball hits the ground.
the vertical component of velocity as the ball hits the ground.
the velocity of the ball as it hits the ground.
Page 17
Caldervale High School
CfE Higher Physics
Often we must split the initial velocity into horizontal and vertical components,
vH and uV.
Worked Example 2
A ball is projected at an angle of 30o the horizontal, with a velocity of 48 ms-1.
Calculate
a)
b)
c)
d)
e)
the horizontal component of the initial velocity.
the vertical component of this initial velocity.
the maximum height.
the time of flight, assuming a flat surface.
the horizontal range of the ball, assuming a flat surface.
Page 18
Caldervale High School
CfE Higher Physics