Forces and Motion 5.5:
Graphical analysis of braking distances, speed, kinetic energy of moving
vehicle
Doc Brown's Physics exam study revision notes
See parts 5.2, 5.6 and 5.8 via
index link below
INDEX of physics notes:
reaction times, stopping distances of road vehicles, Newton's 2nd Law,
KE calculations
5.5 Graphical
analysis of braking distances, speed and kinetic energy of a moving road vehicle
What is the relationship between braking
distance and kinetic energy?
Graph 1a assumes a uniform decrease in velocity i.e. uniform deceleration
See part 5.8
Some advanced
calculations on braking force and removing vehicle kinetic energy
Diagram KEY: KE = kinetic energy (J),
m = mass (kg), u = initial velocity (m/s),
v = final velocity (m/s), s = speed (m/s)
a = acceleration or
deceleration (m/s^{2}), W = work
done (J), F = force (N),
d = distance (m)
The graph 1b above takes the thinking distance, braking
distance and stopping distance data and plots them against the typical speed of a road vehicle.
Obviously, all the distances increase with increase in
speed, but note two other very important points.
You should notice ...
(i) two of the graphs curve upwards, so there is a sort
of 'accelerating' effect of speed on the braking distance and overall
stopping distance (the latter is due to the increase in braking
distance).
Stopping distance and braking
distance are not proportional to speed, and crucially, the
braking distance is proportional to speed squared. This means
the stopping/braking distances increase faster than the increase in
speed.
e.g. doubling speed
quadruples the braking distance (2 ==> 2^{2} = 4) and
trebling speed increases the braking distance nine times (3 ==>
3^{2} = 9).
The thinking distance is
roughly proportional to speed, the graph is ~linear and does not
curve upwards. This is because your response time, if fully alert,
is pretty constant, so if your speed doubles, you just go twice as
far in the same response time.
(ii) and if you examine the graph or data carefully, you
can see that doubling the speed quadruples the braking distance.
This means by doubling your
speed, approximately quadruples the stopping distance, obviously
something you need to bear in mind the faster you drive.
Doubling
speed quadruples braking distance and trebling speed increases it
nine times! (see the REMINDER below)
This is discussed further and is related to the formula
for kinetic energy KE = ½mv^{2}.
By doubling the speed, you quadruple
the kinetic energy of the car, hence you have quadrupled the kinetic
energy to be removed by braking (because
KE
v^{2}). See graphs 2 and 3 and notes below.
Therefore, on doubling the speed,
for a constant braking force, you have four times as much KE to remove
and will need four times the distance to remove it.
For more on kinetic energy
calculations see
Kinetic
energy store calculations
Question to illustrate some of the
ideas above and using the chart below.
When travelling at 20 mph a driver's
thinking distance is 6.0 m and the braking distance is 6.0 m.
(a) What is the stopping distance?
stopping distance = thinking distance
+ braking distance = 6.0 + 6.0 =
12.0 m
(b) Estimate the total stopping distance
at 40 mph (scaling up factor of 2).
If the thinking distance is 6 m at 20
mph, it will be double that at 40 mph, 6 x 40 / 20 = 12 m.
From the KE argument and KE
v2, the braking distance increases by the square of the scale factor.
So the braking distance 6 x 2^{2}
= 24 m
Therefore the stopping distance is 12
+ 24 = 36 m
(check on the chart)
(c) Estimate the total stopping distance
at 80 mph (scale factor 4).
If the thinking distance is 6 m at 20
mph, it will be quadruple that at 40 mph, 6 x 80 / 20 = 24 m
The braking distance increases by the
square of the scale factor.
So the braking distance 6 x 2^{4}
= 96 m
Therefore the stopping distance is 24
+ 96 = 120
m (not on the chart)
INDEX of physics notes on
reaction times, stopping distances of road vehicles, Newton's 2nd Law,
braking friction force, KE calculations
Keywords, phrases and learning objectives for
the physics of road vehicles
 braking distance graphs and calculations
Be able to interpret a graphical analysis of stopping distances in the
context of speed and the kinetic energy of a
moving road vehicle.
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INDEX of physics notes on
reaction times, stopping distances of road vehicles, Newton's 2nd Law,
braking friction force, KE calculations
