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4th form: Motion & Forces
Scalars and Vectors
• Measurable quantities can be divided into
scalars and vectors.
• Scalar quantities have a magnitude (size)
only
– Examples?
• Mass, distance, speed, time...
• Vector quantities have both a magnitude
and a direction associated with them
– Examples?
• Force, displacement, velocity...
Adding scalars and vectors
• Adding scalar quantities is easy, just add
the magnitudes
– eg if it takes 15 minutes to eat lunch and you
have a further 45 minutes before lessons,
how long was the break?
• Adding vectors, we have to take the
direction of the quantities into account
– eg pushing a car against friction
Adding vectors
Adding vectors (more generally)
• The resultant R is found by combining all
the component vectors together
– It is the single vector which is equivalent to
the action of all the component vectors
• This is A-level
stuff...
Speed: a reminder
• Speed is a measure of how quickly
something is moving
distance travelled
speed 
time taken
• Actually, the above formula really tells you
the average speed during the time interval
• As the time interval gets smaller, you get
closer to calculating the instantaneous
speed.
Displacement-time graphs
• Try to describe the motion shown in the graph
– What does the slope of the line represent?
– What does the slope of the dotted line tell you?
Displacement-time graphs
Constant speed forward
Constant speed backwards
Slope=average
speed of return
journey
Slope = speed
Speed=5/0.42=11.9 km/h
stationary
After 160 minutes, we
are back where we
started
Calculating speed
• The slope of the graph gives the speed
(strictly the velocity)
–
distance travelled
slope 
time taken
(a) Slope = 60/10 = 6.0 m/s
(b) Slope = 0/5 = 0.0 m/s
(b)
(a)
(c)
(c) Slope = -100/25 = -4.0 m/s
(d)
(d) Slope = 40/15 = 2.7 m/s
• The steeper the line, the higher the speed
Displacement-time graphs
How would
you
represent
something
getting
slower?
x
t
Note: distance can also become negative, if object travels in the opposite direction
Speed and Velocity
• The velocity of an object gives its
instantaneous speed and direction
– (This is called a VECTOR)
• As with displacement, the sign of the
velocity indicates the direction
– a negative velocity means speed in the
opposite direction
Speed and Velocity
• Going from A
to B: + velocity
• Going from C
to F: - velocity
Velocity-time graphs
• Try to describe the motion shown in the graph
– What does the slope of the line represent?
– Where is the object not moving?
10
8
Velocity (m/s)
6
4
2
Time (s)
0
0
-2
-4
-6
10
20
30
40
Velocity-time
graphs
Constant speed
Constant acceleration
Gradual
slowing
forwards
10
More rapid slowing
8
Velocity (m/s)
6
4
stationary
2
Time (s)
0
0
10
20
30
40
-2
-4
-6
Reversing direction
and speeding up
Constant speed
backwards
Slowing to a
stop
Acceleration
• Acceleration is the rate of change of
velocity
change in velocit y m/s
accelerati on 
s
time
taken
m/s
2
– If you are speeding up, acceleration is +
– If you are slowing down, acceleration is -
Acceleration is the slope of the
velocity graph
Acceleration = (v-u)/t
So for this region:
a= 8/4 = 2 m/s2
and for this region:
a= 0/6 = 0 m/s2 (constant v)
What about the displacement?
• Displacement = velocity × time
• i.e. the area under the
graph
So in the first 4s:
1
distance travelled   8  4
2
 16 m
In the next 6 seconds
distance travelled  6  8
 48 m
so total distance in 10 s  64 m
Tachographs
• A tachograph is an
instrument which records
the velocity-time graph of
a vehicle.
• It is used to check that
EU regulations limiting
the time lorry and bus
drivers can spend at the
wheel are obeyed
– 9 hours/day
– 45 minute break every 4.5
hrs.
Forces: a reminder
• A force is a “push” or a “pull”. Unit: newton (N)
• Forces arise due to the interaction of two (or
more) objects.
• Not all forces require contact, some can act at
a distance
– e.g. gravity, magnetism
• Forces are vectors,
Direction matters
Weight: a reminder
• “Mass” is a measure
of how much “stuff”
an object contains
– measured in kg
• “Weight” is the force
that object exerts due
to the effect of gravity
• So an astronaut has
the same mass on
Earth or the Moon,
but his weight will be
different
– measured in newtons
(N)
On Earth,
W  mg
weight
mass
g ≈10 N/kg
gravitational
field strength
Representing forces
• Forces can be represented with arrows,
whose length indicates the size of the
force.
Force diagrams
• A free body diagram can be very useful to
analyse the forces acting on an object
• We draw it isolated from its surroundings
and show all the forces acting
What forces are acting here?
• Draw on as many as you can think of…
Tension in the rope
Weight and reaction
(for each person)
Push and friction
(for each person)
Combining forces
• If several forces act on an object, we can
work out the equivalent single resultant
force by adding them up, taking direction
into account.
3N
• What is the resultant?
– 4 newtons downwards
6N
• How about these?
4N
1N
5N
4N
4N
0N
4N
6N
7N
Newton’s
1st law
Balanced forces
• It is possible to have all forces balanced, so
the resultant = 0.
For a plane flying
at constant speed
and height:
Thrust = drag
Lift = weight
• In this case, no resultant force acts and the
object continues to move at constant
velocity (or remain stationary if it wasn’t
moving).
Newton’s 1st Law
• A body will remain at rest or, if moving,
continue to move at a constant velocity,
unless acted on by a force.
Unbalanced forces
• If the resultant force is not zero, a net force is
acting on the body and its motion will change.
• It will accelerate in the direction of the force.
• thrust > drag
and
lift > weight,
so aeroplane accelerates and takes off
Newton’s
2nd law
Force causes acceleration
• When a force acts on a body, it changes
its velocity
F  m a
force
mass
acceleration
• If no resultant force acts, there is no
acceleration (Newton’s 1st law)
• Remember, acceleration can mean a
change of speed or direction
• 1 N is the force which accelerates 1 kg at
1 m/s2
F=ma
• So:
– For a given mass, a
bigger force
produces a bigger
acceleration
– For a given force, a
smaller mass
experiences a
bigger acceleration
Force, mass and acceleration
1) A force of 1000 N is applied to push a mass of 500 kg.
How quickly does it accelerate?
2) A force of 3000N acts on a car to make it accelerate by
1.5 m/s2. How heavy is the car?
3) A car accelerates at a rate of 5 m/s2. If it weighs 500 kg
how much driving force is the engine applying?
4) A force of 10 N is applied by a boy while lifting a 20 kg
mass. How much does it accelerate by?
Remember Weight?
• We had W  mg
where W was the weight – the force due to gravity
• Now we know F  ma
so F  W and ma  mg
or a  g
Acceleration
due to gravity
gravitational
field strength
On Earth: g ≈10 N/kg, a ≈10 m/s2
Investigating F, m and a
• We can measure
acceleration with light
gates
• What happens as you
vary:
• Why do we need a
ramp?
• How do we set the
right angle?
– The mass on the
hanger?
– The mass of the
trolley?
What do we find?
• acceleration is
proportional to force
aaF
• acceleration is
inversely proportional
to force
a a 1/m
Counter
force
Horizontal
motion
Driving
force
Driving force – provided by rider/engine
Counter force – air resistance and friction
• Driving force < counter force:
vehicle slows down
• Driving force = counter force:
vehicle moves at constant
velocity
• Driving force > counter force:
vehicle speeds up
Falling Objects
An object falls because of its weight (force
due to gravity)
When object falls freely – no other forces act
on it so resultant force is just its weight.
Remember F = ma?
Acceleration of 10m/s2 is constant for all
objects.
Classic experiment
• So if we dropped a hammer and a feather
at the same time, which would hit the
ground first? Why?
• Hammer & Feather
Drag
• Objects moving in a fluid have drag force.
• For objects travelling through the air we call
this drag force air resistance.
• Air resistance increases with speed.
• So as a falling object speeds up, the
resultant force decreases. This means the
acceleration decreases.
Reaching a constant velocity
Object reaches a constant
velocity when the drag
force/air resistance is
equal & opposite to its
weight.
Resultant force = zero
Acceleration = zero
Velocity = terminal
velocity
Why does a car have a top
speed?
The AR 8C has a 4.7 litre 450 bhp (340 kW)
engine to provide driving force.
Force means acceleration, so why can’t the car
accelerate forever?
What determines terminal
velocity?
• Frontal area
• Shape
• Mass
• Surface
Stages of a parachute jump
Just after letting go...
•Velocity =0
•Drag = 0
•Force = weight
•Acceleration = g
Falling quite fast now...
•Velocity is high
•Drag is large
•Force < weight
•Acceleration < g
Falling at a constant speed...
•Velocity
constant
•Drag = weight
•Force = 0
•Acceleration = 0
Pull the ripcord...
•Velocity still
high
•Drag > weight
•Force upwards
•Acceleration
upwards (so
speed of fall
decreases)
Drift downward...
•Velocity
constant (slower)
•Drag = weight
•Force = 0
•Acceleration = 0
Label the graph
Urban Myth?
• So, would a penny dropped from a
skyscraper kill someone it hit at the
bottom?
• See here for the answer
• (also here if you wonder
about bullets coming down)
Springs: a reminder
• We have seen that springs obey Hooke’s
Law:
– The extension is proportional to the force
applied (up to some limit)
F  kx
Other “stretchy” things
• Hooke’s Law also applies to other objects...
– Metal bars, wires, bones, even glass!
• ...up to a point
– If you go beyond that point you may get failure
(snap) or permanent deformation (doesn’t
return to original shape)
Hooke’s Law limit
Rubber bands
• Rubber bands are elastic, can be
stretched and return to their original
length, BUT they do not obey Hooke’s Law
• How can you tell?
• Describe how it stretches...